Revisiting Spacetrack Report #3

Transcription

AIAA 2006-6753
Revisiting Spacetrack Report #3
David A. Vallado*
Center for Space Standards and Innovation, Colorado Springs, Colorado, 80920
Paul Crawford†
Crawford Communications Ltd., Dundee, DD2 1EW, UK
Richard Hujsak‡
Analytical Graphics, Inc., Exton, PA, 19341
and
T. S. Kelso§
Center for Space Standards and Innovation, Colorado Springs, Colorado, 80920
Over a quarter century ago, the United States Department of Defense (DoD) released the
equations and source code used to predict satellite positions through SpaceTrack Report
Number 3 (STR#3). Because the DoD's two-line element sets (TLEs) were the only source of
orbital data, widely available through NASA, this code became commonplace among users
needing accurate results. However, end users made code changes to correct the
implementation of the equations and to handle rare cases encountered in operations. These
changes migrated into numerous new versions and compiled programs outside the DoD.
Changes made to the original STR#3 code have not been released in a comprehensive form
to the public, so the code available to the public no longer matches the code used by DoD to
produce the TLEs. Fortunately, independent efforts, technical papers, and source code
enabled us to synthesize a non-proprietary version which we believe is up-to-date and
accurate. This paper provides source code, test cases, results, and analysis of a version of
SGP4 theory designed to be highly compatible with recent DoD versions.
T
I.
INTRODUCTION AND HISTORY
he Simplified General Perturbations (SGP) model series began development in the 1960s (Lane 1965), and
became operational in the early 1970s (Lane and Cranford, 1969). The original release of the refined Simplified
General Perturbations-4 (SGP4) propagator source code was Spacetrack Report Number 3 (Hoots and Roehrich,
1980). That release resulted from a user compatibility survey of space surveillance operational sites and official
users. The magnitude of the resulting variations spurred an effort to promote better compatibility for users. The
intent was to get the operational community, as well as ordinary users, synchronized with respect to the
implementation. The best vehicle for this was a technical report, including the computer source code. It was
designed for the widest possible dissemination. Although most of the equations were given, the use of the source
code became common practice for using Two-line Element (TLE) sets.**
*
Technical Program Manager, Center for Space Standards and Innovation, 7150 Campus Dr, Suite 260,
[email protected], AIAA Associate Fellow.
†
Principal Engineer, 25 Blackness Avenue, [email protected].
‡
Orbit Determination Lead Engineer, 220 Valley Creek Blvd, [email protected].
§
Technical Program Manager, Center for Space Standards and Innovation, 7150 Campus Dr, Suite 260,
[email protected], AIAA Associate Fellow.
**
Note that the code is not vetted as a consensus standard. The well-confirmed and long-established industry consensus standards
process requires consensus on all elements of a technique and its implementation throughout a wide community of experts. There
is no formal consensus standard for orbit determination or propagation.
1
American Institute of Aeronautics and Astronautics
Spacetrack Report Number 3 officially introduced five orbital propagation models to the user community—SGP,
SGP4, SDP4, SGP8 and SDP8—all “generally” compatible with the TLE data. At the time, SGP had just been
replaced by SGP4/SDP4 (the latter having included deep-space perturbations). The SGP8/SDP8 model was
developed to alleviate deficiencies of SGP4/SDP4 for the special cases of orbital decay and reentry. The approach
provided a closed-form solution based on the general trends of orbital elements as they neared reentry, and was quite
successful. However, there is no evidence to suggest that SGP8/SDP8 was implemented for operational TLE
formation.
After STR#3, Spacetrack Report Number 6 (Hoots, 1986) was publicly released by North American Aerospace
Defense Command (NORAD). Some researchers initially assumed this release was intended to update portions of
the SDP4 deep-space routines, but the actual intention was to document HANDE* and had little to do with
SGP4/SDP4. Nevertheless, it provided amateur satellite trackers and researchers with a confirmation of identified
deficiencies in the original validation and verification efforts. This report has not been as widely circulated as
STR#3, which benefited from its early electronic availability (Kelso, 1988).
In the early 1990s, the NASA Goddard Space Flight Center (GSFC) obtained a copy of the 1990 standalone
SGP4 code† from project SpaceTrack as part of a study on orbit propagation models for the SeaWiFS Mission (Patt
et al., 1993). In 1996–7 they released the unrestricted code on the Internet and to numerous organizations around the
world involved in the SeaWiFS Mission. It confirmed changes already discovered by many independent researchers,
and we refer to it simply as the “GSFC version.”
In 1998, Hoots published a history of the equations, background, and technical information on SGP4. In 2004,
Hoots et al. published a complete documentation of all the equations (including the deep-space portion). These
publications cover the incorporation of resonances, third-body forces, atmospheric drag, and other perturbations into
the mathematical technique. We note that all published reports on SGP4 have suggested only improvements in the
code used to implement it, and not any changes to the underlying theory. Thus, the equations in Hoots (2004) should
be representative of the current mathematical theory. This is a fundamental and essential assumption we use in this
paper.
Outside the DoD, perhaps the most comprehensive external version of the software resided with Paul Crawford.
His “Dundee code” kept track of the many changes inferred by real-world observations by independent researchers,
and those confirmed by DoD releases. Many of the results contained in the code pre-date matters that were later
confirmed in the DoD standalone releases. We use the change history from the Dundee in this analysis.
A. Motivation
Spacetrack Report Number 3 noted the importance of using the specific equations and data input to ensure
proper operation and we repeat it here. “The most important point to be noted is that not just any prediction model
will suffice… The NORAD element sets must be used with one of the models described in this report in order to
retain maximum prediction accuracy.” The numerous releases and modifications to the original SGP4 standalone
code have made it virtually impossible to satisfy that direction today. For instance, using element sets generated
with the operational SGP4 code will not reproduce the same ephemeris as the original STR#3 code (without
modifications) would. Similarly, using this TLE data in another general perturbations propagator will result in
completely erroneous results. Simply converting the orbital elements to an osculating state vector and propagating
with a numerical propagator is equally invalid. These are consequences of the model-based parameter estimation
technique of orbit determination, and are most noticeable when using general perturbation techniques.
In fact, one may infer that none of the public releases meet this criterion because Kaya, et al. (2004) says “Air
Force Space Command (AFSPC) developed Astrodynamic Standard Software to emulate the operational
astrodynamic algorithms used by the Space Control Center (SCC) in the Cheyenne Mountain Operations Center
(CMOC)” by “extracting desired algorithms from the larger programs in the Space Defense Operations Center
(SPADOC) within the SCC.” Thus, there are multiple versions of the SGP4 code even within the DoD. We must
recognize that the true official code is inextricably linked and embedded within the operational computer system at
CMOC (we designate it as the “operational” version). CMOC uses this operational version to produce all the TLE
data that are distributed daily to worldwide users. A similar “standalone” version of the official code is maintained
*
The HANDE model was intended to replace the analytical SGP4/SDP4 model. It incorporated the effects of the Jacchia
dynamic atmosphere models for the average solar flux during the propagation interval, while retaining the speed and character of
an analytic general perturbations model. It also included the full Brouwer gravity solution, much of which had been dropped for
the SGP4 simplification. The code was implemented in the operational system, but its use is unknown.
†
It appears that the merged SGP4/SDP4 models were now referred to simply as ‘SGP4’ from this 1990 code onwards.
2
by technical offices within AFSPC, which, under various organizational names,* published the Spacetrack series of
reports. The mention of emulating the operational codes leads us to think that AFSPC routinely tests and aligns
these two versions for compatibility. Spacetrack Report Number 3 report contained a snapshot of this standalone
code in 1980 and is the basis for our discussion.
Kaya et al. (2001) note the lack of enforcement for early AFSPC instructions (publicly available administrative
documents) concerning the use of their standalone code, and discusses changes in AFSPC policy about releasing
code. We see this in the evolution of Air Force Space Command Instructions. These documents imply that models
and computer codes have been extracted from larger programs, modified frequently, and that those modifications are
not promulgated or available to the broader user community.†
Perhaps the best motivation for the paper came from a 1998 version of AFSPCI 33-105, which stated,
The need for this instruction was identified by the lack of any HQ AFSPC procedures for releasing a certain set of
software, commonly called the "astrodynamics algorithms," used in the Space Defense Operations Center system
(SPADOC 4C) for the space control mission. With no configuration control in place, various versions of executable and
source code of the "astrodynamics algorithms" have been used for certain contracts and research projects.
1.1. Over the past 15 years or so, various commercial companies have produced and marketed products that these
companies claim contain some of AFSPC's astrodynamics algorithms. Not only are these claims very difficult to confirm,
very few of these claims, if any, have ever been confirmed. Also, in many cases, AFSPC has no documentation that states
why, when, and from whom the contractor obtained the command's code. Consequently, AFSPC and other DoD units
may have purchased their own software, often unknowingly.
1.2. Frequently, the algorithms and code contained in these products were outdated versions or had even been modified
without consultation and certification from AFSPC. Additionally, the contractor rarely provides source code of their
proprietary system to AFSPC so AFSPC cannot confirm whether the system's software actually contains the AFSPC
"astrodynamics algorithms." Consequently, AFSPC cannot perform verification and validation that the astrodynamics
algorithms have been utilized correctly in decision support systems, potentially critical to the space support provided to
other combat units. Because of the severity of the problem with AFSPC's astrodynamics algorithms, an overall
instruction for all of the command's software is required.
Thus today, there are perhaps more versions in use than at the time of original publication and compatibility and
interoperability for users has been impacted. Many organizations routinely use a “version” of SGP4 that they
received from “someone” at “sometime”. Precise documentation is often scarce. Thus, a primary motivation for this
paper is to bring the community up to speed with respect to the current implementation of SGP4 and the TLE data
released by NORAD.
B. Purpose
The technical community has increasingly sought more information about SGP4 because its TLE data set
continues to be widely disseminated even today and represents the only ‘public’ source of data covering the majority
of orbiting objects. Although many of today’s most important operations have switched to numerical processing
methods, the analytical approach still has value, especially when dealing with large numbers of satellites. Examples
of these include:
•
Rapid searches for satellite visibility for ground stations, and generation of communication schedules.
•
Programmed tracking of medium beamwidth antennas (or initial acquisition for narrow beamwidth
auto-track systems) using limited CPU power embedded devices.
•
Investigations into initial orbit design based on low-precision requirements, such as general sensor
and/or ground station visibility statistics.
*
We provide background information on some of the organizational acronyms used within this paper in the Appendix as they
may be confusing.
†
In the late 1990’s AFSPC formalized the STR#3 advice and implemented regulations mandating procedures pertaining to the
use and distribution of the standalone code stating in a 1998 version of AFSPCI 33-105 that, “AFSPCI 60-102, Space
Surveillance Astrodynamics Standards, requires that legacy government astrodynamics software be used in new systems to
ensure interoperability with Space Defense Operations Center system (SPADOC 4C) orbital data and to reduce acquisition costs
by using verified and validated standard astrodynamics algorithms that are Government Off-The Shelf (GOTS) software”. The
2004 version of AFSPCI 33-105, says, “AFSPCI 60-102, Space Surveillance Astrodynamic Standards, mandates that only
standard constants, physical models and astrodynamic algorithms will be used in all AFSPC systems requiring space vehicle
trajectory data from or providing space vehicle trajectory data to the Space Control Center (SCC),” implying that standards at
that time were not “legacy … software.”
3
•
Rapid assessment of close conjunctions (http://CelesTrak.com/SOCRATES) (Kelso and Alfano, 2005)
can be made computationally efficient by pre-processing with analytical techniques, and then applying
numerical techniques only to those cases that appear to warrant additional consideration.
This paper provides source code, test cases, results, and analysis of a version of SGP4 designed to be similar to
the standalone code. Because the complete equations for SGP4/SDP4 are given in Hoots et al. (2004), they are not
repeated here. Instead, the focus is more on the actual code development, testing methodology, and results. The
references at the end of the paper attempt to list the various papers that document the SGP4 theory and practice. This
will establish a consistent new baseline and permit improved accuracy of operations for worldwide users that
routinely process TLE data. The TLE are routinely available from CelesTrak (http://CelesTrak.com) and AFSPC
(www.space-track.org). The basic format for the data has not changed much over the years and is described in many
places and we have included a discussion in the Appendix.
II.
PROGRAM INTERFACE ISSUES
A few technical questions and comments are necessary to effectively integrate these analytical solutions into
today’s environment.
A. Theoretical Issues
TLE data support a mix of coordinate systems and analytical theories. The SGP theory was largely based on
Kozai (1959), while the SGP4 theory was primarily based on Brouwer (1959). The two theories are rather different,
but both are still in use today. Neither the Kozai nor Brouwer theory originally included drag effects, so different
treatments of atmospheric drag are in use. SGP approximates drag via rate changes of mean motion (Hilton and
Kuhlman, 1966), while SGP4 uses power density functions (Lane and Cranford, 1969; Lane and Hoots, 1979) that
require a term that encapsulates the ballistic coefficient, Bstar (see Vallado, 2004: 113–116). Simplified force
modeling and the batch-least-squares processing of observational data often yield a Bstar that has “soaked up” force
model errors. Occasionally, one finds negative Bstar values, indicating erroneously that energy is being added to the
system, but this is simply a consequence of the limited SGP4 force modeling with respect to the actual dynamical
environment.
B. Configuration Control
TLE data do not reveal which version of SGP4 was employed to estimate the orbital parameters. Different
definitions of the so-called True Equator Mean Equinox (TEME) coordinate system and time systems may also have
been used at different times. Without a list of dates to synchronize these changes with historical TLE data, the user
must decide which version of the SGP4 propagator might be consistent. Because the accuracy of the propagator is
generally in the kilometer-level range (Hartman, 1993), this may not be a problem for most cases, but as we’ll see
shortly, some of the technical modifications can cause results to differ by hundreds of kilometers. This topic is
perhaps the least likely to have a simple solution, but could potentially account for significant differences in
ephemeris generation.
C. Data Formats
The TLE format appears to have changed slightly over the years, and numerous TLE data were disseminated
with missing or erroneous values. Some of these cases simply test the error handling of the code and its ability to
handle premature ending of the propagation.
The TLE Element Type is always set to zero for distributed data, although STR#3 suggests the following
assignments: 1 = SGP, 2 = SGP4, 3 = SDP4, 4 = SGP8, 5 = SDP8. The TLE sets also use differing formats (e.g., use
of leading zeros, or not). Sometimes, parameters are omitted within the TLE data (e.g., a second time derivative of
mean motion or Bstar drag term equal to zero). These variations can confound fixed-read implementations in a
computer program. The parsing of the TLE files is a bigger problem in languages such as C where the fixed-position
approach (common in FORTRAN) is unusual, and where the ‘NUL’ (zeroth in the ASCII collating sequence) has a
special end-of-string significance. Additionally, there are possible differences between DOS-formatted text files
(CR/LF for end of line) and UNIX format (LF only). Attention is paid in the conversion utility to account for these
discrepancies and the parsing routine is kept separate from the SGP4 routines to permit users the option of tailoring
their parsing needs for a particular operation.
The TLE format has a simplistic form of error checking by having a checksum character for each line; however,
it is prudent to check for other ‘fixed’ aspects (such as the “1” and “2” for each line, matching satellite numbers on
the two lines, variable ranges, etc.) since the modulo-10 checksum only provides a 90% detection rate for uniformly
4
random errors. Even with the checksum, there has been some ambiguity over the value assigned to the characters
(the + sign in particular, which we believe should be zero), some additional explanation can be found on the web.*
D. Coordinate System
The actual SGP4 model has little need for any specific coordinate or time system (e.g., the near-Earth part is
rotationally symmetric about the pole), but when used for propagating TLE generated by DoD it becomes important
to use the same coordinate system as the DoD orbit determination routines use. The commonly accepted output
coordinate system is that of the “true equator, mean equinox” (TEME) (Herrick, 1971:325, 338, 341). An exact
operational definition of TEME is very difficult to find in the literature, but conceptually its primary direction is
related to the “uniform equinox” (Seidelmann, 1992:116, and Atkinson and Sadler, 1951). The intent was to provide
an efficient, if approximate, coordinate system for use with the AFSPC analytical theories. Technically, the direction
of the uniform equinox resides along the true equator “between” the origin of the intermediate Pseudo Earth Fixed
(PEF) and True of Date (TOD) frames (Vallado, 2004:211, 221). It is found by observing that vGAST82 may be
separated into its components. Thus,
K
K
rTOD = ROT 3(−θ GAST 82 )rPEF and θ GAST 82 = θ GMST 82 + Eqe82
K
K
rTEME = ROT 3(−θ GMST 82 )rPEF
K
K
rPEF = ROT 3(θ GMST 82 )rTEME
(1)
We recommend converting TEME to a truly standard coordinate frame before interfacing with other external
programs. The preferred approach is to rotate to PEF using Greenwich Mean Sidereal Time (GMST), and then rotate
to other standard coordinate frames. Conversions are well documented from this point. To implement, you simply
apply a sidereal rotation about the Z-axis by GMST (using UT1 as we discuss later). Because polar motion has been
historically neglected for General Perturbation (GP) applications, we assume that the pseudo Earth-fixed frame is
the closest conventional frame.†
If a rotation is made to TOD using the equation of the equinoxes, several approximations are introduced with the
calculation of the nutation of the longitude (∆W) and the obliquity of the ecliptic (e). There are at least three possible
sources of uncertainty with this method: the number of terms to include in the nutation series, the inclusion of the
post-1996 "kinematic correction" terms to the equation of the equinoxes, and small angle approximations. After
choosing the length of the IAU 1980 nutation series (4, 10, and 106 terms are popular choices with 4 being most
common), the transformation is sometimes further reduced by assuming that ∆W ≈ 0, e ≈ ε , and ∆e ≈ 0. This results
in a nutation matrix that is significantly simpler than the complete nutation matrix, although the complete form is
more common today. The equation of the equinox may be approximated by ignoring the "kinematic correction"
terms starting in 1997 [such that EQeqe1980 ≈ ∆WCOS(e)]. Finally, because some of the multiplicative quantities are
small, second-order terms may be neglected.
However, you should be aware of an additional nuance, specifically the ‘of date’ and ‘of epoch’ formulations.
• TEME of Date—With this option, the epoch of the TEME frame is always the same as the epoch of the
associated ephemeris generation time. The transformation to ECEF is done by first finding the
conversion from TEME to TOD (third equation in Eq. (1)). Next the standard transformation from TOD
to ECF is computed. We could have gone directly to PEF without the TOD frame (second equation in
Eq. (1)), but this implementation enables comparison with the TEME of Epoch approach. All
transformations are found using the complete IAU-76/FK5 formulae, including nutation.
• TEME of Epoch—In this approach, the epoch of the TEME frame is held constant. Subsequent rotation
matrices must therefore account for the change in precession and nutation from the epoch of the TEME
frame to the epoch of the transformation. This is accomplished by finding a static transformation from
TEME to J2000—this includes the equation of the equinoxes, the nutation, and the precession which are
all calculated at the epoch of the TLE. This static transformation is applied at each time requested in an
*
The data available on CelesTrak undergoes extensive testing prior to publication. This includes checksum, individual column
checking (e.g., a number field can only have 0 – 9, a decimal field only a period), and range checking, where appropriate (e.g.,
inclination between 0 and 180). Not all archive sources of TLE have had such checks performed, and end users are advised to
consider this aspect before using those TLE. Additional information can be found at the CelesTrak website.
http://celestrak.com/NORAD/documentation/checksum.asp
†
We assume that CMOC orbit detemination approximates the reference frames of radar and optical differently, and that
numerical and analytical orbit determination methods use different techniques due to the differences in TEME, ECI, and the
uncertain use of polar motion in coordinate systems.
5
ephemeris generation. Once the J2000 vector is found, standard techniques can convert this to other
coordinate systems, at the appropriate time. This is computationally intensive, and introduces error into
the subsequent solutions. All transformations, after the initial static calculations, are computed using the
complete IAU-76/FK5 formulae, including all terms of the nutation theory.
Researchers generally believe the ‘of date’ option is correct, but confirmation from official sources is uncertain,
and others infer that the ‘of epoch’ is correct. To be complete, we provide the equations and an example problem of
both in the Appendix.
E. Time System Issues
Time accounting within SGP4 is referenced to the epoch of the TLE data. This practice makes individual
satellite ephemeris generation and use relatively easy, although it can complicate multiple satellite analyses. The
time system is assumed here to be UTC, but no formal documentation exists and UTC, as currently defined, was
only introduced in 1972. UT1 is needed to calculate GMST for the coordinate transformations discussed in the
appendix, but it is unknown whether UT1 or UTC is what is required by the software, although we assume UT1 for
this paper. The error associated with approximating UT1 with UTC is within the theoretical uncertainty of the SGP4
theory itself. Except for the GMST calculation, this paper and code assumes time to be realized as UTC.
Time accounting also affects how the year of epoch values are handled within a system. This feature is only
peripherally related to SGP4, and not part of the mathematical definition. It appears in the epoch calculations and
affects how the two-digit year of the TLE is treated. Several possibilities exist. If the year is less than 50, 57, or
some other value, one can add 2000, otherwise, 1900 is added. Of course, these are only temporary fixes with the
correct option to be the use of a 4-digit year, Julian Date, Modified Julian Date, etc. During the so-called "Y2K"
millennial rollover, some attention was focused here although nothing apparently changed.
It is doubtful that a leap-second capability was implemented into the peripheral software for SGP4 since the
historical source code uses relative “time since epoch”. Any such addition is clearly outside the mathematical
formulation of SGP4, but necessary for programs to interface with other agencies. As some software libraries have
no support for the ‘61 second’ minute that is needed to properly represent or convert UTC time at the point of leapsecond insertion, we suspect this is simply ignored for the majority of non-critical users, and a 1-second timing
difference will occur sometime during the period between TLE updates where the leap second is added or
subtracted. Although this is outside the direct scope of SGP4, it is part of many System Acceptance Tests for large
programs, and is included here as a reminder of those operations.
F. GHA Calculation
The Greenwich Hour Angle is usually calculated using the Julian Date. However, you can also find expressions
using the elapsed time from some epoch. Among the versions of SGP4 that are available today, several epochs arise:
1950 Jan 1 0h, 1970 Jan 0 0h, and 1970 Jan 1 12h, UT1. The various combined constants illustrate the potential for
error when using this approach. As new timing systems are developed, the associated timing parameters change
slightly. The precision of these parameters also change slightly. Consider the following examples from various
versions:
Jan 1, 1950 0 hr (original STR#3)
THETA = 1.72944494D0 + 6.3003880987D0*DS50
Jan 0, 1970 0 hr
C1
= 1.72027916940703639D-2
THGR70 = 1.7321343856509374D0
FK5R = 5.07551419432269442D-15
C1P2P = C1+TWOPI
THGR = DMOD(THGR70+C1*DS70+C1P2P*TFRAC+TS70*TS70*FK5R,twopi)
These approaches yield “essentially” the same values. A series of calculations were constructed to test these
against the IAU convention (Vallado, 2004:191) using the Julian centuries of UT1 (TUT1).
2
−6 3
θ GMST1982 = 67,310.548 41s + (876,600 h + 8,640,184.812 866 s )TUT1 + 0.093104TUT
1 − 6.2 x10 TUT 1
(2)
The results showed comparisons of about 10-9 degrees difference. This is well below the level that the answers
would be affected. We have chosen to implement the conventional approach of Eq (2).
III.
COMPUTER CODE DEVELOPMENT
The revised computer code developed in this paper is provided in C++, FORTRAN, MATLAB, and Pascal to
permit reasonable flexibility for applications (C++ is given in the appendix as this language is becoming
6
commonplace). Conversion to other languages should be aided by the re-structuring effort that has been performed
on the code.
There can be large variations between the numerous implementations of SGP4—hence the need to establish a
newer baseline that is compatible with CMOC as closely as possible to provide enhanced compatibility. Where
obvious updates and corrections have been made and verified, we account for each in our revised code. For other
improvements that appeared “obvious” to us, we tried to determine if these changes might be present in today’s
standalone version.
Our starting point was the 1980 version from STR#3. From this point, STR#6, the Dundee modifications, and the
GSFC code release verified several suspected code changes. There were too many changes in this update to describe
them all—we list a few of the major ones below. Note that all satellite numbers refer to examples in the test case
file (sgp4-all.tle). Also, all element plots are osculating values.
• A primary change from STR#3 was the merging of SGP4 and SDP4 code. A large number of
researchers had noticed the commonality of the two models and simplified the code in this manner,
however, not all had recognized and simplified the initialization code. Due to this simplification, most
now refer to the merged SGP4/SDP4 models simply as ‘SGP4’.
• Although ultimately STR#6 has little relevance for TLE use, one notable change was the move to
double-precision code throughout (rather than the mix of single and double in the original) and
corresponding increase in accuracy for certain astrodynamic constants, all made practical by the
improvement in computing power since STR#3. Such changes do not improve the “accuracy” of the
model as such, but they lead to “smoother” behavior which helps with some tasks (such as differential
correction), and to greater consistency of results on differing computer systems and/or compilers.
• Solving Kepler’s equation was updated, but not completely fixed. The solution of Kepler’s equation
continues to present challenges in astrodynamics hundreds of years after its introduction. The original
1980 version of SGP4 had a fixed limit of 10 iterations and a tolerance of 10-6, but contained no code to
prevent certain high-eccentricity orbits from failing to converge. Spacetrack Report Number 6 changed
the tolerance to the tolerance to 10-12 (commensurate with double-precision work) and the GSFC
version tried to solve the convergence problem by removing the iteration limit. However, these are
incomplete approaches and can still result in infinite loops. The revised version code includes an
updated SGP4 routine following the Dundee version that allows realistic controls on the iterations.
Figure 1 shows the impact of this practice.
• The practice of only computing the lunar-solar terms if propagation time changes by more than 30
minutes to save CPU effort was dropped, thus resulting in smoother behavior for deep-space orbits with
small time steps. This was the only function of the SAVTSN variable in the original DPPER subroutine.
This resulted in ‘choppy’ behavior in some ephemerides from the STR#3 version.
• The application of periodic lunar-solar perturbations was updated. There are actually three problems
relating to the application of the periodic lunar-solar perturbations. The first of these, sometimes known
as the “Lyddane bug” (because it was first noted in independent investigations of the Lyddane
modifications in DPPER), is due to the jump in the actan/atan2 output where the perturbed value due to
the discontinuity of this function at either 90°/270° or ±180° (respectively). In the STR#3 code, the
actan discontinuity occurred at 270°. Spacetrack Report Number 6 tried using the atan2 function, but
that simply moved the discontinuity to 180°. The GSFC code (the IF statements at the end of the ‘apply
periodics’ section in DPPER) confirmed the suspicions of several researchers about the need to evaluate
the relative quadrant of the resulting angle and to correct accordingly. A similar problem exists with the
modulo 2π reduction of the XNODE variable. The effect of not correcting the quadrant is illustrated in
Fig. 2. This problem also occurs when intrinsic functions (mod, atan, etc.) are used instead of the
STR#3 versions. We feel intrinsic functions are better suited for the program, but that full envelope
testing of the Lyddane implementation is probably in order.
• The second difficulty with the lunar-solar perturbations was the initialization of deep-space terms based
on perturbed values. This was corrected in the DPPER and SGP4 routines of the Dundee and GSFC
versions. In STR#3, the terms computed during initialization assumed fixed epoch values for
inclination, etc., but of course they are perturbed by the deep-space terms. The approach used by the
Dundee and GSFC versions includes any terms based on the Keplerian orbit being re-computed based
on the new perturbed values.
7
30.32
Corrected
30.30
Inclination (deg)
30.28
30.26
STR#3
30.24
30.22
30.20
30.18
0
200
400
600
800
1000
1200
1400
1600
Min from Epoch
360
STR#3
Mean Anomaly (deg)
300
240
180
120
60
Corrected
0
0
200
400
600
800
1000
1200
1400
1600
Min from Epoch
Figure 1. Solving Kepler’s Equation for Satellite 23333. The mean anomaly (bottom) illustrates
the severe discrepancy in incorrectly solving Kepler’s equation after about 200 minutes. The effect
also shows up in the inclination (top). The problem existed in the STR#3 version, shown here, but
corrections were attempted in STR#6 and the GSFC version, with a better approach in the Dundee
version. The inclination plot also shows the choppy (but smaller) behavior of the 30 minute
updating of the lunar-solar terms in the STR#3 version before about 200 minutes. Notice that the
effect goes away after about 1400 minutes.
8
278.5
STR#3
278.0
Argument of Perigee (deg)
277.5
277.0
276.5
276.0
Corrected
275.5
275.0
274.5
250
300
350
400
450
500
550
600
650
700
750
Min from Epoch
30000
25000
y-component
Position Components (km)
20000
STR#3
15000
10000
Corrected
5000
z-component
0
-5000
-10000
x-component
-15000
-20000
250
300
350
400
450
500
550
600
650
700
750
Min from Epoch
Figure 2. Lunar-solar modifications for Satellite 23599. The argument of perigee and positional
components illustrate the discrepancy in incorrectly updating the lunar-solar perturbations, and not
accounting for the proper quadrant in the periodic calculations. The problem existed in the STR#3
version, shown here, and an attempted correction was made in STR#6, but it was mostly corrected
in the GSFC version.
•
The third area of confusion with the lunar-solar perturbations is the decision for when to use the
Lyddane modification, and we refer to it as the “Lyddane choice” (Satellites 14128, 20413). Lyddane
(1963) reformulated the Brouwer expansions (done in Delaunay variables) in Poincare variables. Both
are canonical, and the Poincare variables were intended to be non-singular for small eccentricity and
inclination values. Because this was a reformulation, its use was intended for all computations, and
9
remains that way today in the Navy Position Partials and Time (PPT3) (Hoots et al., 2004). During the
development of SDP4 equations, some SIN(inclination) divisor problems were noted and the Lyddane
formulation was examined. Because it also exhibited singularities, an alternate formulation was sought,
ultimately resulting in new parameter choices. The decision was made to use these variables when the
inclination was less than 11.4592º (0.2 rad).
Thus, the code implementation introduced two methods of applying the lunar-solar perturbations in the deepspace code, with the Lyddane modification used with smaller inclinations to avoid a divide-by-zero type of
computation problem. In the STR#3 version, the choice was based on the unperturbed epoch inclination proximity to
11.4592º. In STR#6 (and the GSFC code subroutine DPPER), the test used the perturbed inclination (XIP, with the
secular term applied, unlike the XQNCL common term used in STR#3). This approach leads to a potential for the
model switching lunar-solar methods as a function of propagation time, which is clearly undesirable. Note that the
difference is usually small and relies on positional differences rather than the actual positional values. However, the
results can be greatly magnified in some orbits (satellite 20413 which is a multi-day orbit). The basic effect can be
demonstrated by satellite 14128 after about 2000 minutes from epoch when the perturbed inclination emerges above
11.4592º as shown in Fig. 3.
2.0
Position Component Difference (km)
1.5
1.0
y-component
0.5
0.0
-0.5
z-component
-1.0
-1.5
-2.0
0.0
500.0
1000.0
1500.0
2000.0
2500.0
3000.0
Min from Epoch
Figure 3. Lyddane Choice Modification for Satellite 14128. The difference in positional
components illustrates the discrepancy in applying the Lyddane modification using the perturbed
inclination rather than the original inclination. The effect exists when comparing the GSFC and
STR#3 versions. Note the differences diminish once the inclination crosses the 11.4592º threshold
(after 2000 min).
A few comments are necessary on the Lyddane choice. This case occurs exceedingly infrequently as the satellite
inclination must be very close to the 11.4592º limit, and it must be a deep-space satellite. Without carefully crafted
test cases, this (like several of the problems we discuss) is not easy to detect in normal operation. At the time of
coding the STR#3 version, this form of software testing was not a commonly taught practice.* We can consider
several methods of resolving the situation, such as (a) going back to the STR#3 practice of testing the unperturbed
epoch inclination at t = 0 and using that as a fixed decision for the TLE, (b) testing the perturbed inclination at each
propagation time (as in the GSFC version), or (c) making more significant code changes to find a smoother way of
*
In 1980, the limitations of computer memory and storage space often dictated stringent code length requirements. Code that
would only be exercised once or twice, if a small effect, could be safely omitted in deference to more critical techniques,
applicable to numerous satellites. The testing philosophy of the time also influenced the outcome. Using a single set of test cases
for all analyses was quite common. The notion of targeted test cases for individual loops and constructs in the code itself didn’t
arise until many years later. Thus, this small nuance could easily have been missed by the testing of the time, or by code
limitations themselves.
10
blending the two lunar-solar perturbation methods. Although a rare case, we think some fix should be included and
hope that AFSPC will confirm the current state of the code so users can be compatible in all cases. For the present
paper, we have included Option (b) in the code with qualifiers to assist in location and potential future resolution.
However, we note that there is probably a better “crossover” point to apply the Lyddane modification (Option c) that
will not result in such large discrepancies in the two ephemerides, but time did not permit a thorough investigation
and recommendation for such a change.
Next, the following changes were made to comply with modern programming standards, and to facilitate any
changes in the future. With the exception of the variable precision and the integrator issues (discussed later), none of
these changes affected the technical performance of the program and could be considered “cosmetic”.
• Implicit typing in FORTRAN was replaced by comprehensive variable declarations. This was a critical step
before conversion to C++ and others. Modern compilers can generally sort out the variable names, but the
possibility of mistaken variables, variables being set to zero and used in calculations, etc., was too great. In
addition, knowing which variables were calculated and set assisted the process of forming structures.
Finally, this also eliminated much of the need for the FORTRAN SAVE command to hold values between
function calls with certain compilers.
• Structures were created to pass the large amounts of data between functions. Numerous variables were
passed between functions in the original code. With no typing in the original code, this approach proved
relatively easy, but it was difficult to gain an understanding of the underlying structure. The structures were
set up to support integrated near-earth and deep-space functionality provided in the code. This change also
supported processing multiple satellites at one time. While processing a single satellite is illustrative for
simple scenarios, it is unrealistic for many modern applications. For instance, the SOCRATES effort uses
TLE data to generate potential conjunction information. During these runs, one must have two or more
satellites in memory at one time.
• GOTO statements in FORTRAN were eliminated, using more modern constructs. This old programming
construct is often seen in legacy programs, but completely unnecessary with modern programming
techniques and tools. Looping and decision constructs were inserted, as appropriate.
• Intrinsic functions replace user-written routines. Trigonometric and exponential routines should use
intrinsic calls within the programming language. The only exception should be in cases where a specific
quadrant, ordering, etc. is required. None of these were deemed necessary within the SGP4 routines.
• Initialization functions were separated for better organization. The code was modularized, keeping
initialization functions separate from routine function use. Although modern compilers can generally sort
these differences out, the code is easier to maintain if the functions are isolated for a particular operation.
The reorganization of the computer code simplified the processing flow. In addition, simple timing studies
performed during the original development demonstrated increased processing speed of about 10%. The
basic program structure is illustrated in Fig. 4.
• Variable names were changed to better conform to the variables they represent. Many variable names were
limited to conform to the former FORTRAN limitation of six (6) characters. This is no longer necessary
and has been dropped. Variable names were changed to match “standard” nomenclature, such as that used
throughout Vallado (2004). Constants were kept as constants in the code, and not assigned as variables with
limited precision.
11
SGP4
if
metho
d
if
method
if init
if
initds
DSPACE
DPPER
if init
INITL
SREZ
Near
Earth
SREZ
DSCOM
if <225
Deep
Space
One call each time
DPPER
if init
DSPACE
SREZ
DSCOM
Initialization
integrated
if
initds
SREZ
SGP4
SGP4init
INITL
Near
Earth
if
metho
d
DSPACE
if
metho
d
DPPER
if <225
DSCOM
Deep
Space
DPPER
One initialization call
Routine calls toSGP4
DSINIT
Figure 4. SGP4 Structural Organization. The computer program structure is shown for the
original and derivative programs (top) and the revised version for this paper (bottom). Note that
the initialization was interspersed throughout the original program, while it is better isolated in the
revised code.
12
IV.
SAMPLE TEST CASES
The original STR#3 included several test cases and sample outputs, but only for two sample satellites. Given the
number of branches possible in the deep-space case, many more tests are needed to fully test the code. The original
cases have been extended over the years as users have encountered real-world situations. The only other official test
cases are referenced in AFSPCI 60-102. No publishing information is given in the AFSPCI.
Because the theory is based on analytical expressions, comparisons are relatively simple because the output
should be the same from each program. Different programming languages (C++, FORTRAN, MATLAB, or Pascal)
and compilers produced very small differences, but these were well below the accuracy of two-line element sets that
are commonly used, and below the comparison between differing implementations.
For analysis, the computer code was set up with three primary execution paths. First, there is a “verification”
path in which the program accepts an input TLE file that includes start and stop dates and time steps. Mechanizing
this step was important to quickly review any changes against “known” test results. The second mode processed the
entire space object catalog from one day before to one day after the epoch time. The negative time propagation was
chosen to highlight any difficulties in the secular integrator part of the deep-space code—a most convoluted
example of programming in the official versions. Several space object catalogs (having about 9000 satellites in
them) were tested from the historical database. This provided a quick-look at performance for each of the programs
against a wide range of satellite orbits. The third mode of operation is the standard mode whereby input element sets
are read, and some operation takes place with the data. We separated the driver and TLE-conversion function from
the SGP4 code, to permit a user to modify the driver as needed, without having to change the underlying SGP4 code.
The test cases were divided into two categories. First, there were verification runs that tested the basic algorithm
implementation. The second set of tests demonstrates cases that we believe indicate additional technical
considerations that AFSPC may have incorporated in their models, or should consider in the future.
V.
VERIFICATION TEST CASES
Essentially, these cases allowed several features of SGP4 to be tested, but the answers were generally agreed
upon during the testing phase of research for this paper. Cases for which there were technical questions about how
the code was implemented are discussed in a subsequent section. The element sets were sorted numerically in the
computer file to aid location of specific test cases, but are grouped here by effect. Comments were added to indicate
what each test was accomplishing. The original SGP4 model had two types defined in the code, normal (near Earth)
and ‘simplified drag’, while the original SDP4 had three types, normal (deep space), resonant (12h Molniya style)
and synchronous (24h GEO). Table 1 shows a sample. The file (sgp4-ver.tle) is on the Internet at the web site listed
at the end of the paper, and in the Appendix.
Table 1. SGP4 Verification Test Cases. These satellites highlight the primary test cases used for
analysis and verification of the SGP4 code. A few other satellites are included in the full test set.
The satellites used for the figures are also included, but at a reduced ephemeris density. The file
gives the applicable time range in minutes from epoch (MFE). The original STR#3 tests are kept
for continuity.*
Satellite
Category
Comments
00005
Near Earth
TEME example satellite.
28129
Deep Space
A GPS navigation satellite in a near circular 12h orbit.
26975
Resonant
Molniya style debris launch. Exercises the 0.5 to 0.65 eccentricity branches in deep space.
08195
Resonant
Molniya launch. Exercises the 0.65 to 0.7 eccentricity branches of the deep-space code.
09880
Resonant
Molniya launch. Exercises the 0.7 to 0.715 eccentricity branches of the deep-space code.
21897
Resonant
Molniya launch. Exercises the eccentricity branches above 0.715, with a negative Bstar value.
22674
Resonant
Rocket body, similar to 21897 (e > 0.715) but positive Bstar
28626
Synchronous
Low-inclination (< 3 deg) geostationary orbit that shows the problems in premature correction
of negative inclination at around 1130 minutes from epoch.
25954
Synchronous
Low-inclination GEO case like 28626, shows negative inclination problem at around 274
*
All TLE data given in this paper is representative of actual satellites and can be obtained from www.CelesTrak.com except for
the original Report #3 test cases which do not appear in the archives.
13
Satellite
Category
Comments
minutes from epoch.
24208
Synchronous
Geostationary orbit above 3 deg.
09998
Synchronous
Relatively high eccentricity for GEO (e = 0.027) shows secular integrator problem clearly.
14128,
04632
Synchronous
Geostationary orbit close to 0.2 radian inclination. Shows Lyddane choice problem at about
2080 minutes and about –5000 minutes from epoch.
20413
Deep Space
Long period orbit (~4 days) shows Lyddane choice at 1860 minutes from epoch.
23333
Deep Space
Very high eccentricity, shows Kepler solution problems in Report #3 code.
28623
Deep Space
Deep-space object with low perigee (135.75 km) that uses the branch (perigee < 156 km) for
modifying the ‘s4’ drag coefficient.
16925
Deep Space
Deep-space object with very low perigee (82.48 km) that uses the second branch (perigee <
98 km) for limiting the ‘s4’ drag coefficient to 20
06251
Near Earth
Near Earth normal drag case. The perigee of 377.26 km is low, but above the threshold of 220
km for simplified equations, so moderate drag case.
28057
Near Earth
Near Earth normal drag case but with low eccentricity (0.000 088 4) so certain drag terms are
set to zero to avoid math errors / loss of precision.
29238
NE/S
Near Earth with perigee 212.24 km, thus uses simplified drag branch (perigee < 220 km) test.
28350
NE/S
Near Earth low perigee (127.20 km) that uses the branch (perigee < 156 km) for modifying
the ‘s4’ drag coefficient. Propagation beyond approximately 1460 minutes should result in
error trap (modified eccentricity too low).
22312
NE/S
Near Earth with very low perigee (86.98 km) that uses the second branch (perigee < 98 km)
for limiting the ‘s4’ drag coefficient to 20. Propagation beyond approximately 2840 min
should result in error trap (modified eccentricity too low).
28872
NE/S
Sub-orbital case (perigee –51 km, lost about 50 minutes from epoch) used to test error
handling.
23177,
23599
Deep Space
Lyddane bug at less than 70 min and 380 min respectively, with atan2(), but no quadrant fix
26900
Deep Space
Lyddane bug at 37,606 min, negative inclination at 9313 min
29141
Near Earth
Last stages of decay. Crashes before 440 min
11801/
88888
Deep Space,
Near Earth
Original STR#3 report test cases
VI.
EXPECTED CODE UPDATES
Although we searched many locations to obtain the latest openly available documentation on official AFSPC
practice, a few topics remain unknown. The primary areas of discussion are those giving the largest differences in
results—specifically negative inclinations, integrator problems, and solution of Kepler’s equation. If the reader is
aware of other corrections, we would appreciate learning about them. The intention is to produce a new baseline that
is as close as possible to the current operational version to enhance compatibility for the external user. While we
could not verify these, we felt the changes were so obvious that AFSPC has already made them, thus we have
included the options in the code. We used a comment (keyword “sgp4fix” in the codes) by each change to make any
future retraction or addition easier. For official users who are constrained by the AFSPCI 33-105 restrictions and
have only an executable version of the current code, it should be a simple matter to confirm these fixes.*
*
The AFSPC instructions have applied to different entities over time. By August 2004, AFSPCI 33-105 states the instruction
“applies to Headquarters Air Force Space Command (HQ AFSPC), subordinate units, supporting activities and contractors who
develop, acquire, maintain or deliver computer software, including all systems that require astrodynamic algorithms. It also
applies to Air Force Reserve Command (AFRC) and Air National Guard (ANG) units gained by HQ AFSPC.” Previous versions
incrementally added each of these groups, so it would appear that the scope of the intended audience is increasing with time.
14
A. Error Checking
We increased the amount of error checking in our code to handle cases such as decayed satellites, or satellites
having inconsistent values. CelesTrak employs a significant amount of error checking on the TLE data, but
programs allowing the user to enter data could result in values that would cause errors. Inclination values near 180.0
degrees can cause divide-by-zero problems in the initialization and the routine operation. This is fixed by setting a
tolerance in both routines. The decay condition simply checks the position magnitude on each step.
B. Constants
Kaya et al. (2001, 2004) focuses on the difficulties encountered when mixing WGS-72 and WGS-84 constants.
Because the SGP4 codes contain references to WGS-72, AFSPC may have updated the constants to WGS-84, but
there is no other documentation supporting this so we present the development in case new official documentation is
released. However, because many operational sites may still have embedded software containing a version of SGP4
using WGS-72, and the fact that the accuracy of the theory would not really be impacted, AFSPC may well have
chosen to retain the older set of constants to better maintain interoperability with its internal resources. We use
WGS-72 as the default value. As with other changes we discuss, this is only necessary to interface with external
programs, but it will cause a difference in ephemeris results. The proper sequence to form the constants for WGS-72
is shown below. Note that we determined µ from the SGP4 code value of XKE because it is not specified directly in
the code, and this makes future revisions easier. We also provide TUMin because XKE is simply the reciprocal of
this quantity. TUMin is possibly more familiar as it is the number of minutes in one time unit—a necessary
conversion when using canonical constants.
Table 2. WGS-72 Constants. The fundamental and derived constants are shown below. Notice
that XKE and TUMin are reciprocal values. The original STR#3 listed XKE as 0.074 366 916.
Symbol
Calculation
Value
398,600.8 km3/s2
µ
6378.135 km
RK
J2
0.001 082 616
J3
–0.000 002 538 81
J4
–0.000 001 655 97
XKE
0.074 366 916 133 17 /min
60/sqrt(RK3/µ)
TUMin
13.446 839 696 959 31 min
sqrt(RK3 /µ)/60
If we use WGS-84 values, we find the following values.
Table 3. WGS-84 Constants. The fundamental and derived constants are shown below. The
zonal harmonic values are converted from the normalized values.
Symbol
Calculation
Value
398,600.5 km3/s2
µ
6378.137 km
RK
J2
C2,0 = –0.000 484 166 850 00 0.001 082 629 989 05
J3
C3,0 = 0.000 000 957 063 90 –0.000 002 532 153 06
J4
C4,0 = 0.000 000 536 995 87 –0.000 001 610 987 61
XKE
0.074 366 853 168 71 /min
60/sqrt(RK3/µ)
3
TUMin
13.446 851 082 044 98 min
sqrt(RK /µ)/60
Other constants may not be familiar at first. For example, XPDOTP is a conversion from rev/day to rad/min.
XPDOTP = 1440.0/2π = 229.183 118 052 329 3.
RPTIM is simply the rotational velocity of the earth in rad/min. Note this does not use the GRS-80 defining
parameter for the rotation of the Earth, 2*π / (86,400/1.002 737 909 350 795) * 60.0 = 7.292 115 855 3x10-5, but
rather the GRS-67 value that Aoki et al. (1982) used in the definition of time.
RPTIM = 7.292 115 146 7x10-5 * 60.0 = 0.004 375 269 088 02 rad/min.
Other constants are combined with other values, or use the values mentioned previously in their formulation. We
do not believe any update has occurred to any of the embedded constants in the deep-space portions as no
documentation has ever suggested this.
15
C. Negative Inclination Orbits (Satellite 25954, 28626)
Deep-space orbits with low-inclination values (typically geosynchronous orbits) can, due to the effects of lunar
and solar gravity, result in a negative inclination with time. This can create a step-function discontinuity in the
positional components. Normally this is resolved by shifting the ascending node longitude by 180°. In the computer
code, we corrected this by removing the quadrant check from DSINIT before the ‘initialize resonance terms’
section, but kept the check in SGP4 before the ‘long period periodics’ section.
Satellites 25954 (at times beyond 274 minutes) and 28626 (at times beyond 1130 minutes) illustrate the effect of
correcting negative inclination prematurely. The bottom graph in Fig. 5 of z-position reveals a discontinuity around
this time in incorrect implementations of the code.
0.0190
Corrected
0.0185
0.0180
Inclination (deg)
0.0175
GSFC
0.0170
0.0165
0.0160
0.0155
0.0150
0.0145
0.0140
-1440
-1200
-960
-720
-480
-240
0
240
480
720
960
1200
1440
Min from Epoch
20
15
Corrected
z-Position Component (km)
10
5
0
GSFC
-5
-10
-15
-20
-1440
-1200
-960
-720
-480
-240
0
240
480
720
960
1200
1440
Min from Epoch
Figure 5. Negative Inclination Performance for Satellite 25954. The inclination and zcomponent of the position vector show the step function discontinuity of the previous SGP4
versions. The STR#3 and GSFC versions exhibits the problem while the Dundee version does not.
16
D. Integrator Problems (Satellite 09880)
The original FORTRAN codes contained a generous mix of GOTOs and other structures that made accurate
debugging nearly impossible. One area that appears to have suffered from this practice was the secular integrator
used for 12h and 24h resonant cases. In particular, several satellites show difficulties when propagated ‘backwards,’
that is going to some time away from epoch (either positive or negative time) and then taking time steps towards the
epoch again. The problem seems to be with the setup of the positive and negative steps (stepp and stepn) with values
of 720 minutes in the DSPACE routine (SREZ in the older programs). It appears that ‘cleaning up’ the code has
fixed the problem. The original STR#3 style of logic would integrate from epoch to the required time using a Taylor
series approximation:
F (x + h ) = F (x ) +
h
h2
F ′(x ) + F ′′(x ) + "
1!
2!
(3)
where the integral at epoch F(0) is defined as zero. If the time (h) from epoch was greater than 720 minutes, it would
step in 720 minute intervals (x = 720, 1440, …), recalculating the 1st and 2nd derivatives each time and saving the
current (multiple of 720 minute) values for future use. Integrator resets occurred only when crossing the epoch. This
was correct and efficient provided the model was only called with increasing time steps (either positive or negative),
but it gives inconsistent results if you go to a time far from the epoch and return ‘backwards’ towards the epoch. The
code from this paper always integrates from the epoch to the required time, and restarts each time the model is called
with a ‘backwards’ step. This is slightly less CPU efficient but leads to repeatable results.* Satellite 09998
demonstrates this symptom quite clearly as it is propagated from one day before the epoch until one day after the
epoch, though all of the resonant and synchronous cases show it to some extent. Consider Fig. 6.
E. Solving Kepler’s Equation (Satellite 23333)
The partial fix discussed earlier handles a majority of the problem cases one would encounter in operations.
However, additional robustness could be handled via several alternative methods. A simple but very effective fix for
this is covered in Crawford (1995) where it is noted that the difference between mean and eccentric anomaly is
never more than ±e radians, so if you limit the first Newton-Raphson correction to somewhere around this, it
converges reliably for all cases. As this problem only applies to very high eccentricity orbits, an even simpler option
fixes a limit of 0.9 – 1.0 for the maximum correction. Another option is that of Nijenhuis (1991), who examines the
problem for eccentricities of 0.999 and 0.9999 and also examines the overall CPU load as well as the iteration count.
Note that this iteration is not the ‘traditional’ iteration to find eccentric anomaly discussed in the literature (e.g.,
Vallado, 2004: 72–85). Results for this change were shown in Fig. 2.
A series of tests were run to determine the number of iterations for a complete satellite catalog, and the
satellite tests we have included with this paper. For an example case of e = 0.9 the STR#3 version took an average of
5.685 iterations with a maximum of 8. The corrected Dundee version had an average of 3.984 iterations, with a
maximum of 5. As with the Lyddane choice mentioned earlier, this change affects only a very small number of
satellites.
*
The ProjectPluto code had a variation on this method. It always integrates in the “shortest path” (improving CPU
use slightly over both the STR#3 logic and our ‘repeatable’ logic) but did not keep to the 720 minute step size for
re-computing terms, leading to discrepancies.
17
38221.70
38221.68
38221.66
Semimajor Axis (km)
38221.64
38221.62
38221.60
corrected
38221.58
38221.56
gsfc
38221.54
38221.52
38221.50
-1440
-1320
-1200
-1080
-960
-840
-720
-600
Min from Epoch
0.25
Position Component Difference (km)
0.20
0.15
0.10
x-component
0.05
0.00
z-component
-0.05
y-component
-0.10
-1440
-1320
-1200
-1080
-960
-840
-720
-600
Min from Epoch
Figure 6. Propagation Problems for Satellite 09998. The integrator problem is shown by
looking at the semimajor axis (top) and the positional component differences (bottom). The scale
is small, but the semimajor axis clearly shows the jump caused by incorrect integrator
performance near 720 minutes prior to the epoch. The problem appears in all older versions.
VII.
COMPARISON ANALYSES
Many versions of SGP4 are available in code today, although most are initially from STR#3. Virtually none have
been re-worked to restructure the code or to provide multiple computer programming languages and test results. Our
aim is to correct that situation. Note that the basic structure of the computer code given in this paper has been
available for several years in FORTRAN, Pascal, Ada, and C++ on the following web site
(http://CelesTrak.com/software/vallado-sw.asp), although there has been extensive analysis to update the code for
this paper. There are only three known “official” versions with which we could make comparisons. These include:
18
STR#3
GSFC
JPL
(FORTRAN)
Both the original single/double mix, and a double-precision version (just by adding the IMPLICIT DOUBLE
statement) of STR#3 code were used. The electronic code was released to all users who asked for it. T. S.
Kelso released an electronic package of the 1980 report in December 1988.
(FORTRAN)
http://seawifs.gsfc.nasa.gov/SEAWIFS/SOFTWARE/src/bobdays/sgp4sub.f (original)
Note this version is no longer available at this website although numerous downloads are known by
organizations and countries. In addition, the code is still easily found on archive pages throughout the
internet. A current site is similar, but potentially confusing as the subroutine name is the same, but the
module is clearly labeled as a Brouwer–Lyddane model.
http://www.icess.ucsb.edu/seawifs/seadas/src/utils/bobdays/sgp4sub.f (new, but different file)
(FORTRAN)
ftp://naif.jpl.nasa.gov/pub/naif/toolkit/FORTRAN/PC_Linux/packages/toolkit.tar.Z
An additional source of SGP4 implementations is the JPL NAIF ‘spicelib’ toolkit, with source files ev2lin.f
(basically SGP4.FOR equivalent), dpspce.f (basically SDP4.FOR) and zznrddp.f (basically the DEEP.FOR).
A few other codes were examined to determine what other researchers had done with the code. Examples that were
tested but not included in the results presented here were:
ProjectPluto (C++)
http://www.projectpluto.com/sat_code.htm
Not an official version, but it is one of the more interesting and intelligent conversions to C++ available.
TrakStar (Pascal)
http://CelesTrak.com/software/tskelso-sw.asp
This is a very well known example, but is essentially a direct conversion of STR#3 and so it can be expected
to behave in a manner similar to the STR#3 double-precision case.
Dundee (C)
http://www.sat.dundee.ac.uk/~psc/sgp4.html
The original translation into C by Paul Crawford and Andrew Brooks was virtually identical in behavior to
STR#3, but with much better code structuring. Then many of the other fixes included such as the Kepler’s
equation solution and secular integrator were added. The update over the last year for this paper had all of the
corrections discussed and agreed with the authors, resulting in virtually identical results to the paper’s
versions.
Because our version of SGP4 does not claim to be the official version, it was important to compare the results
over a wide range of test conditions, and to compare with the released official versions. Specifically, the verification
and stressing test cases provided a technical look at the performance, but these comparison tests were intended to
show the robustness of the calculations under full-catalog simulations. Tests were run on several complete catalogs
for varying dates. Each satellite was propagated from –1440 minutes to 1440 minutes at 20-minute time steps. The
results were then compared between programs. The C++, FORTRAN, MATLAB, and Pascal versions gave virtually
the same results, as shown in Fig. 7.
19
100000
10000
1000
Delta r (m)
100
10
1
0.1
0.01
0.001
0.0001
0.00001
0
200
400
600
800
1000
1200
1400
1600
1000
1200
1400
1600
Period (min)
100000
10000
1000
100
Delta r (m)
10
1
0.1
0.01
0.001
0.0001
0.00001
0
200
400
600
800
Period (min)
Figure 7. SGP4 Full-Catalog Comparisons. Maximum differences between ephemerides
generated for two days are shown. Note the small scale for the C++ and FORTRAN comparison
(top). The Pascal comparison (bottom) shows very small additional variations and these are from
the 8-byte versus 10-byte precision in the language.
Comparisons were then run between the versions. Each figure shows the largest difference between the
simulations, and each satellite is plotted against the orbital period. The scales are kept constant within each figure to
permit rapid assessment of the differences. Figure 8 shows the results compared to the GSFC version. This was
important to illustrate the similarity with the last known release.
20
100000
10000
1000
100
Delta r (m)
10
1
0.1
0.01
0.001
0.0001
0.00001
0
200
400
600
800
1000
1200
1400
1600
1000
1200
1400
1600
Period (min)
100000
10000
1000
100
Delta r (m)
10
1
0.1
0.01
0.001
0.0001
0.00001
0
200
400
600
800
Period (min)
generated for 2 days are shown. The top plot shows the paper C++ version against the GSFC code,
while the bottom plot shows the comparison to the GSFC code, but only for propagations positive
from the epoch. The differences are all related to the integrator problems before 720 minutes prior
to epoch with geosynchronous and semi-synchronous orbits.
As Fig. 8 shows, the GSFC version is very close to our revised version, and nearly identical to the performance
between languages for the revised versions. The minor differences (usually a few meters) in the resonant cases (718minute Molnyia and 1436-minute geostationary orbits) only show up with time steps that ‘go backwards’ in time (a
problem in the secular integrator). In these tests, we begin at –1440 minutes and then step towards zero, before
going ‘forwards’ towards +1440 minutes. In our revised version, the direction of propagation is not important. The
21
GSFC version also has larger errors with the direct/Lyddane choice, and the inclination going negative during
propagation, but these are not shown in Fig. 8 (it requires rare or ‘difficult’ TLEs to show up). Those differences
were discussed earlier.
The comparisons with results from STR#3 show significantly larger differences for almost all satellites. Note
that both (mixed) single and double-precision results are given in Fig. 9.
100000
10000
1000
100
Delta r (m)
10
1
0.1
0.01
0.001
0.0001
0.00001
0
200
400
600
800
1000
1200
1400
1600
1000
1200
1400
1600
Period (min)
100000
10000
1000
100
Delta r (m)
10
1
0.1
0.01
0.001
0.0001
0.00001
0
200
400
600
800
Period (min)
generated for 2 days are shown. The top plot shows the single-precision STR#3 version against the
paper C++ version, while the bottom plot shows the STR#3 double-precision version comparison.
22
Both versions of STR#3 (single/double and double only) show similar results, with agreement to reasonable
accuracy (sub-km) for near-Earth orbits (limited by the precision of specifying the astrodynamic constants). This is
less in deep space, where some of the limited precision (e.g., Kepler’s equation tolerance), the re-computing of
perturbed terms, and the possibly the ‘Lyddane bug’ behavior show up more strongly. The differences between the
GSFC version and the STR#3 versions are nearly identical to those in Fig. 9 for this catalog snapshot. This is
important because many “correct” implementations of SGP4 in use are based on the STR#3 version (e.g., TrakStar),
but this comparison shows the typical additional errors that users can expect as compared to the standalone AFSPC
code.
Despite a rigorous attempt to review the fundamental constructs of the code, the JPL case is not particularly good
in its original form. For near-Earth satellites, there is clearly some problem with the implementation (drag equations
perhaps?) as it is much worse than STR#3 code in mixed precision. The deep-space cases have other problems, one
of which is the choice to zero the LS offsets at epoch* (which does not appear to be correctly implemented in any
case). It also shares the negative inclination problem. These are unfortunate, as it is an interesting attempt to order
and modernize the FORTRAN code, showing some insight into improving things, but missing others (such as the
commonality of the SGP4/SDP4 codes) completely. The Project Pluto code (not shown) compared favorably to the
revised code version but its lack of ‘official heritage’ made it difficult to accept changes based solely on its presence
in this version. Results for the JPL code are shown in Fig. 10.
*
It appears that JPL made the same change as several other authors who assumed that the zeroing of the Lunar-Solar
perturbations at epoch mentioned in STR#6 and the GSFC code also applied to the code when used in the deep space part of the
merged SGP4 model. This is not the case, and the reader is reminded that what matters most for accuracy, if the theory is not
completely documented, is to use the same code for propagating the elements as was used in generating them.
23
100000
10000
1000
100
Delta r (m)
10
1
0.1
0.01
0.001
0.0001
0.00001
0
200
400
600
800
1000
1200
1400
1600
1000
1200
1400
1600
Period (min)
100000
10000
1000
Delta r (m)
100
10
1
0.1
0.01
0.001
0.0001
0.00001
0
200
400
600
800
Period (min)
generated for 2 days are shown. The plot shows the paper C++ version against the original JPL
code (plot on the top). Note that by changing the DOPERT variable in the JPL code, the results
can be improved by about two orders of magnitude (plot on the bottom).
VIII.
AVAILABILITY
The primary computer source code discussed in this paper has been available on the Internet for nearly six years,
but was re-worked for this paper with inputs from many people and organizations. The current code is available in
C++, FORTRAN, MATLAB, and Pascal as these appear to be the most common languages for operations today.
24
The appendices contain definitions and examples of TLE data and TEME conversions, the C++ code, along with the
input TLE data, and results. The code for debugging the software is not included as it was not pertinent to the
discussion. The files have ‘include’ statements which are commented out where these lines of code would be
inserted. All the necessary files are located on the Internet for convenience. They are available from the Center for
Space website:*
http://www.centerforspace.com/downloads/
IX.
CONCLUSIONS
This paper has re-examined the Spacetrack Report Number 3 formulation of analytical propagation. By
incorporating changes posted over the last quarter century, a unified and improved version is presented for general
use. Structural changes to the code have been completed permitting the ability to process multiple satellites at one
time. We chose to omit the Lyddane choice change for certain inclinations to maintain as close a performance to
what we believe AFSPC is doing today. However, we also included comments in the source code to facilitate
location of any updates now, or at a future time. Test cases are included to demonstrate verification of operation
with the branches in the code, for difficult orbits, as well as cases encountered throughout the years. The results
show that continued use of the STR#3 version, and to a lesser extent some of the more recent versions, can result in
potentially large errors when producing ephemerides. We also noted the difficulty with aligning a particular version
of SGP4 with a particular TLE as the data formats and processing have changed throughout the years. Finally, we
hope this form of documentation will motivate similar efforts for additional analytical theories in a similar fashion,
along with satellite data to use with each theory. Any questions, comments, additions, etc. may be addressed to
David Vallado at [email protected].
Acknowledgements
Many people were involved with this project in addition to the co-authors listed. I am very grateful for all the
help, and support I received during this long project. Felix Hoots provided significant insight and details of the
original development and John Seago provided suggestions for a more concise description of the time and
coordinate systems. A special thank you is due to Jeff Beck who provided the MATLAB version based on the C++
code.
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information.
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Springs, CO.
Aoki, S. et al. 1982. The New Definition of Universal Time. Astronomy and Astrophysics. Vol. 105: 359–361.
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_______ 1961b. Appendix to Theoretical Evaluation of Atmospheric Drag Effects in the Motion of an Artificial Satellite.
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California.
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*
Users of Analytical Graphics Inc. Satellite Toolkit (STK) will find the source code integrated within the latest release of the
program.
25
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m-daily Model. Paper AIAA-96-3606 presented at the AIAA/ AAS Astrodynamics Conference. San Diego, CA.
Coffey, S. L., and H. L. Neal. 1998. “An Operational Special-Perturbations-Based Catalog.” Paper AAS 98-113 presented at
the AAS/AIAA Space Flight Mechanics Conference. Monterey, CA.
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Glover, R. A. 1996. “The Naval Space Command (NAVSPACECOM) PPT3 Orbit Model.” NAVSPACECOM Technical
Report.
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U-3871, 17–28.
Hoots, Felix R. 1980. “A Short, Efficient Analytical Satellite Theory.” AIAA Paper No. 80-1659.
_______. 1981. Theory of the Motion of an Artificial Earth Satellite. Celestial Mechanics. Vol. 23, pp. 307–363.
_______. 1986. “Spacetrack Report #6: Models for Propagation of Space Command Element Sets.” Space Command, United
States Air Force, CO.
_______. 1998. “A History of Analytical Orbit Modeling in the United States Space Surveillance System.” Third US-Russian
Space Surveillance Workshop. Washington, D.C.
Hoots, Felix R. et al. 1986. “Improved General Perturbations Prediction Capability.” Air Force Space Command,
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AAS/AIAA Paper No. 83-395.
_______. 1987. An Analytical Satellite Theory using Gravity and a Dynamic Atmosphere. Celestial Mechanics. Vol. 40, pp.
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U.S. Air Force Aerospace Defense Command, Colorado Springs, CO.
Hoots, Felix R., P. W. Schumacher, and R. A. Glover. 2004. History of Analytical Orbit Modeling in the U. S. Space
Surveillance System. Journal of Guidance, Control, and Dynamics. 27(2):174–185.
Hujsak, R. S. 1979. “Spacetrack Report #1: A Restricted Four Body Solution for Resonating Satellites Without Drag.” U.S.
Air Force Aerospace Defense Command, Colorado Springs, CO.
_______. 1979. “A Restricted Four Body Solution for Resonating Satellites with an Oblate Earth.” AIAA Paper No. 79-136.
Hujsak, R. S., and F. R. Hoots. 1977. “Deep Space Perturbations Ephemeris Generation. Aerospace Defense Command
Space Computational Center Program Documentation, DCD 8, Section 3, 82:104.”
_______. 1982. “Deep Space Perturbations Ephemeris Generation.” NORAD Technical Publication TP-SCC 008. 129–145.
Jacchia, L. G. 1970. “New Static Models of the Thermosphere and Exosphere with Empirical Temperature Profiles.” SAO
Report 313. Cambridge, MA: Smithsonian Institution Astrophysical Observatory.
Kaya, Denise, et al. 2004. “AFSPC Astrodynamic Standard Software.” Paper AAS 04-124 presented at the AAS/AIAA
Space Flight Mechanics Conference. Maui, HI.
Kaya, Denise, et al. 2001. “AFSPC Astrodynamic Standards – The Way of The Future.” Paper presented at the MIT/LL
Conference. Lexington, MA.
Kelso,
T.S.
2004.
“Frequently
Asked
Questions:
Two-Line
Element
Set
Format.”
(See
http://CelesTrak.com/columns/v04n03/)
Kelso, T. S., and S. Alfano. 2005. “Satellite Orbital Conjunction Reports Assessing Threatening Encounters in Space
(SOCRATES).” Paper AAS 05-124 presented at the AAS/AIAA Space Flight Mechanics Conference. Copper Mountain, CO.
Kozai, Y. 1959. The Motion of a Close Earth Satellite. Astronomical Journal. Vol. 64, No. 1274, pp. 367–377.
Lane, M. H. 1965. “The Development of an Artificial Satellite Theory Using Power-Law Atmospheric Density
Representation.” AIAA Paper 65-35.
Lane, M. H., and K. H. Cranford. 1969. “An Improved Analytical Drag Theory for the Artificial Satellite Problem.” AIAA
Paper No. 69-925.
Lane, M. H., P. M. Fitzpatrick, and J. J. Murphy. 1962. “Spacetrack Report #APGC-TDR-62-15: On the Representation of
Air Density in Satellite Deceleration Equations by Power Functions with Integral Exponents.” Air Force Systems Command,
Eglin AFB, FL.
Lane, M. H., and F. R. Hoots. 1979. “Spacetrack Report #2: General Perturbations Theories Derived from the 1965 Lane
Drag Theory.” Aerospace Defense Command, Peterson AFB, CO.
Lyddane, R. H. 1963. Small Eccentricities or Inclinations in the Brouwer Theory of the Artificial Satellite. Astronomical
Journal. Vol. 68, No. 8, 1963, pp. 555–558.
Morris, Robert F., and Timothy P. Payne. 1993. “SGP4 Version 3.01 Validation Test Cases.” Publishing data unknown.
Referenced in AFSPC I 60-102.
Nijenhuis, Albert. 1991. Solving Kepler's equation with high efficiency and accuracy. Celestial Mechanics and Dynamical
Astronomy. Vol. 51, No. 4, pp 319–330.
26
Patt, Frederick S., Hoisington, Charles M., Gregg, Watson W., and Coronado, Patrick L. 1993. NASA Technical
Memorandum 104566, Vol. 11 “Volume 11, Analysis of Selected Orbit Propagation Models for the SeaWiFS Mission” available
at http://library.gsfc.nasa.gov/Databases/Gtrs/Data/TM-1993-104566v11.pdf.
Schumacher, P. W., and R. A. Glover. 1995. “Analytical Orbit Model for U.S. Naval Space Surveillance: An Overview.”
Paper AAS 95-427 presented at the AIAA/AAS Astrodynamics Specialist Conference. Halifax, Canada.
Seago, John, and David Vallado. 2000. “Coordinate Frames of the U.S. Space Object Catalogs.” Paper AIAA 2000-4025
presented at the AIAA/AAS Astrodynamics Specialist Conference. Denver, CO.
Tanygin, Sergei, and James R. Wright. 2004. “Removal of Arbitrary Discontinuities in Atmospheric Density Modeling.”
Paper AAS 04-176 presented at the AAS/AIAA Space Flight Mechanics Conference. Maui, HI.
Vallado, David A. 1999. “Joint Astrodynamic Working Group Meeting Minutes.” September 20, 1999. USSPACECOM/AN.
Colorado Springs, CO.
_______. 2001. “A Summary of the AIAA Astrodynamic Standards Effort.” Paper AAS 01-429 presented at the AIAA/AAS
Astrodynamics Specialist Conference. Quebec City, Canada.
_______. 2004. Fundamentals of Astrodynamics and Applications. Second Edition, second printing. Microcosm, El Segundo,
CA.
_______. 2005. “An Analysis of State Vector Propagation using Differing Flight Dynamics Programs.” Paper AAS 05-199
presented at the AAS/AIAA Space Flight Mechanics Conference. Copper Mountain, CO.
Vallado, David, and Salvatore Alfano. 1999. “A Future Look at Space Surveillance and Operations.” Paper AAS 99-113
presented at the AAS/AIAA Space Flight Mechanics Conference. Breckenridge, CO.
27
Appendices
A. Organizational Nomenclature
29
B. Two-Line Element Set Format
30
C. TEME Coordinate System
32
D. Computer Code Listing
35
E. Test Case Listing
37
F. Test Case Results Listing
47
28
Appendix A – Organizational Nomenclature
Tracing the reports, documents, and files back to the original release may present some confusion to users that
are not familiar with the various organizational structures that have been in place over time. We have tried to use the
appropriate organizational names when referencing information, but the following information may help associating
those references. We use DoD, AFSPC, NORAD, CMOC, etc. The following quotes were assembled from the
Cheyenne Mountain website: https://www.cheyennemountain.af.mil.
“The original North American Air Defense Command (NORAD) Combat Operations Center … has evolved into the
Cheyenne Mountain Operations Center (CMOC). The original requirement for an operations center in Cheyenne
Mountain was to provide command and control in support of the air defense mission against the Soviet manned bomber
threat ... In the early 1960s, the advent of an Intercontinental Ballistic Missile (ICBM) attack against North America
became a top priority. Missile warning and air sovereignty were the primary missions in the Mountain throughout the
1960s and 70s. During a brief period in the mid 1970s, the Ballistic Missile Defense Center was installed within the
Mountain. … In 1979, the Air Force established a Space Defense Operations Center [SPADOC] to counter the emerging
Soviet’s anti-satellite threat … The evolution continued into the 1980s when Air Force Space Command [AFSPC] was
created and tasked with the Air Force Space mission … In April 1981, Space Defense Operations Center crews and their
worldwide sensors, under the direction of Air Defense Command [ADC], supported the first flight of the space shuttle …
Oct. 1, 2002 marked the welcoming of two new commands, U.S. Northern Command and U.S. Strategic Command, to
Cheyenne Mountain. CMOC is responsible for providing support to USNORTHCOM's mission of homeland defense and
USSTRATCOM's mission of space and missile warning, formerly associated with U.S. Space Command.
Today, the Cheyenne Mountain Complex is known as Cheyenne Mountain Air Force Station (CMAFS). CMAFS is host
to four commands: North American Aerospace Defense Command (NORAD), United States Northern Command
(USNORTHCOM), United States Strategic Command (USSTRATCOM), and Air Force Space Command (AFSPC).
CMOC serves as the command center for both NORAD and USNORTHCOM. It is the central collection and
coordination center for a worldwide system of satellites, radars, and sensors that provide early warning of any missile,
air, or space threat to North America. Supporting the NORAD mission, CMOC provides warning of ballistic missile or
air attacks against North America, assists the air sovereignty mission for the U.S. and Canada, and if necessary, serves as
the focal point for air defense operations to counter enemy bombers or cruise missiles. In addition, CMOC also provides
theater ballistic missile warning for U.S. and allied forces. In support of USSTRATCOM, CMOC provides a day-to-day
picture of precisely what is in space and where it is located. Space control operations include protection, prevention, and
negation functions supported by the surveillance of space. … Operations are conducted in seven centers manned 24 hours
a day, 365 days a year. The centers are the Air Warning Center, Missile Correlation Center, Space Control Center
[JSPOC], Operational Intelligence Watch, Systems Center, Weather Center, and the Command Center ... The Joint Space
Operations Center (JSPOC) supports United States Strategic Command (USSTRATCOM) missions of surveillance and
protection of U.S. assets in space. The JSPOC’s primary objective in performing the surveillance mission is to detect,
track, identify, and catalog all man-made objects orbiting earth. … The JSPOC maintains a current computerized catalog
of all orbiting man-made objects, charts preset positions, plots future orbital paths, and forecasts times and general
location for significant man-made objects reentering the Earth’s atmosphere.”
The organizational names identify whether they are joint (international), or not. For generic applications, we use
DoD because this is the broadest identification for all the organizations. We generally use NORAD when referring
to the TLE data because their formation was begun under NORAD. Today, the JPSOC [a.k.a. Space Control Center]
within the CMOC produces the TLE data, but we retain the historical name due to its familiarity in the community.
Regulations and documentation have generally come from AFSPC and are identified as such.
29
Appendix B – Two Line Element Set Format
1 6 6 0 9 U
8 6 0 1 7 A
9 3 3 5 2
.
5 3 5 0 2 9 3 4
Right Ascension of
the Node (deg)
Eccentricity
Mean motion derivative
(rev/day /2)
S
.
0 0 0 0 7 8 8 9
Arg of Perigee
(deg)
Mean motion
second derivative
(rev/day2 /6)
S.
Bstar (/ER)
S.
S E
0 0 0 0 0 - 0
1 0 5 2 9 - 3
Mean Anomaly
(deg)
Mean Motion (rev/day)
0
Elem
num
Chk Sum
Yr
Inclination (deg)
Epoch
Day of Year (plus fraction)
Eph
International
Designator
Year Lch# Piece
34 2
Epoch
Rev
Chk
1
Satellite
Number
Class
Card #
The format for the TLE is shown in Fig. 11 with sample data.
.
2
1 6 6 0 9
5 1
.
6 1 9 0
1 3
.
3 3 4 0
0 0 0 5 7 7 0
1 0 2
.
5 6 8 0
2 5 7
.
5 9 5 0
1 5
.
5 9 1 1 4 0 7 0 44786 9
Figure 11. Two-line Element Set Format. An example TLE is shown, with descriptions and
units of each field. Note that the eccentricity, mean motion second derivative, and Bstar have
implied decimal points before the first numerical value. The mean motion derivative is already
divided by 2, and the second derivative is already divided by 6. Shaded cells do not contain data.
The signs may be blank, “+” or “–”. A classification field is sometimes included after the satellite
number.
There are several notes.
1. The maximum accuracy for a TLE is limited by the number of decimal places in each field (Vallado,
2004:116). In general, TLE data is accurate to about a kilometer or so at epoch and it quickly degrades (Hartman,
1993). We note that the SGP4 theory is capable of much better accuracy through additional modeling and sufficient
observational data. Cefola and McClain (1987) noted that certain low-inclination geosynchronous orbits exhibited
large discrepancies from numerical simulations due to oversimplifications in the node rate calculations. Cefola and
Fonte (1996) showed that the addition of additional terms to the theory could improve the overall accuracy by
almost an order of magnitude. Cappellucci (2005) showed that using numerically generated state vectors and
performing an SGP4 orbit determination on the resulting ephemerides produced errors representative of a numerical
technique. This is not unexpected as additional (continuous) observations provide the needed observability over a
simple “3-obs per pass” approach (Vallado and Alfano, 1999). However, the results diverge very rapidly once
outside the fit span of the orbit determination. We do not address orbit determination here.
It is also worth noting that there are numerous other analytical orbital theories that have been developed for a
wide range of applications, but unfortunately, no comprehensive source of data for those theories exists. SGP4 is
interesting because it tries to satisfy many orbital regimes. The Russians developed a series of analytical
propagators, each tuned for a specific satellite regime. The narrower focus permits additional attention to detail, and
higher resulting accuracy. It is hoped that some of these analytical routines can eventually be documented for
general use in the same manner as this paper.
2. The satellite number consists of any numeric value 0 – 99999. Discussions have hinted at a lengthening of the
field size to 7 or 9 characters to accommodate future satellites.
3. Sometimes additional assignments are made – signs = 0; minus signs = 1.
4. The International designator is broken up into the last two digits of the lunch year, the launch number for that
year (3 digits), and the piece of the launch (3 digits). Kelso (2004) also indicates:
[The] International Designator of the object [ ] is an additional unique designation assigned by the World Data Center-A
for Rockets and Satellites (WDC-A-R&S) in accordance with international treaty (1975 Convention on Registration of
Objects Launched into Outer Space). The WDC-A-R&S works together with NORAD and NASA's National Space
Science Data Center (NSSDC) in maintaining this registry. Although there have been some changes in format since it
was first used back in the late 1950s (see "Space Surveillance" in Satellite Times Volume 4 Number 1), the International
Designator indicates the year of the launch (field 1.4 only gives the last two digits), the launch of that year (field 1.5), and
the piece of that launch (field 1.6) for each object. These three fields can be left blank, but all must be present if any is.
Finally, field 1.6 can be either right or left justified—the latter is preferred.
As an aside, there are some significant differences between NORAD's Catalog Number and the International Designator.
For example, NORAD assigns a catalog number based upon when the object was first observed, whereas the
30
International Designator is always tied to the original launch. For example, the 81st launch of 1968 carried four payloads
into orbit: OV2-5, ERS 21 and 28, and LES 6. Together with the Titan 3C transtage rocket body, these objects were
assigned International Designators 1968-081A through E and Catalog Numbers 03428 through 03431. Just this past
October, however, NORAD cataloged two additional pieces associated with this launch as Catalog Numbers 25000 and
25001—they have the International Designators 1968-081F and G.
5. The mean motion rates are not used by SGP4 and are only valid for the older SGP model.
6. Bstar is an SGP4 drag-like coefficient. Usually, ballistic coefficients (BC) are used in aerodynamic theory.
The BC is m/cDA, or the reciprocal (A is cross-sectional area, cD is the coefficient of drag, and m is mass). Bstar is an
adjusted value of BC using the reference value of atmospheric density, ρo = 2.461 x 10-5 kg/m2, at one Earth radius.
BC = Re ρo / (2Bstar)
(B-1)
7. The Ephemeris type is not used external to CMOC. All TLE data is generated by SGP4.
8. From Kelso (2004):
The element set number. Normally, this number is incremented each time a new element set is generated. In practice,
however, this doesn't always happen. When operations switch between the primary and backup Space Control Centers,
sometimes the element set numbers get out of sync, with some numbers being reused and others skipped. Unfortunately,
this makes it difficult to tell if you have all the element sets for a particular object.”
The last column on each line represents a modulo-10 checksum of the data on that line. To calculate the checksum,
simply add the values of all the numbers on each line—ignoring all letters, spaces, periods, and plus signs—and
assigning a value of 1 to all minus signs. The checksum is the last digit of that sum. Although this is a very simple errorchecking procedure, it should catch 90 percent of all errors. However, many errors can still sneak through. To eliminate
these, all data posted on the CelesTrak WWW site not only pass the checksum test, but must also pass both format and
range-checking tests (as described in this article).
The final field on line 2, prior to the checksum, is the rev number. Since there are several conventions for determining rev
numbers, this field also bears some clarification. In NORAD's convention, a revolution begins when the satellite is at the
ascending node of its orbit and a revolution is the period between successive ascending nodes. The period from launch to
the first ascending node is considered to be Rev 0 and Rev 1 begins when the first ascending node is reached. Since many
element sets are generated with epochs that place the satellite near its ascending node, it is important to note whether the
satellite has reached the ascending node when calculating subsequent rev numbers.
31
Appendix C – TEME Coordinate System
This section describes the equations necessary to implement the nutation equations for the TEME approach.
There are two approaches – using the GMST, and using the equation of the equinoxes.
For sidereal time, GMST is needed. GMST is found using UT1. From McCarthy (1992:30)
2
−6 3
θ GMST1982 = 67,310.548 41s + (876,600 h + 8,640,184.812 866 s )TUT 1 + 0.093 104TUT
1 − 6.2 x10 TUT 1
(C-1)
The transformation to ITRF is found using the polar motion (xp, yp) values and the GMST. Note that “PEF”
implies the pseudo-Earth-fixed frame, where polar motion has not yet been applied (Vallado, 2004: 217).
[ W ] ITRF − PEF = ROT 1( y p ) ROT 2( x p )
K
K
rITRF = [ W ]T [ ROT 3(θ GMST 1982 )]T rTEME
K
K
K
K
v ITRF = [ W ]T [ ROT 3(θ GMST 1982 )]T vTEME + ω ⊕ × rPEF
{
}
(C-2)
If the equation of the equinox approach is taken, you must find the nutation parameters. The IAU-80 nutation
uses so-called Delaunay variables and coefficients to calculate nutation in longitude (∆ψ1980) and nutation in the
obliquity of the ecliptic (∆ε1980). (McCarthy, 1992:32)
2
3
M ( = 134.962 981 39° + 1,717,915,922.6330" TTT + 31.31TTT
+ 0.064TTT
2
3
M O = 357.527 723 33° + 129,596,581.2240" TTT − 0.577TTT
+ 0.012TTT
3
µ ( = 93.271 910 28° + 1,739,527,263.1370" TTT − 13.257TTT2 − 0.011TTT
2
3
DO = 297.850 363 06° + 1,602,961,601.3280" TTT − 6.891TTT
+ 0.019TTT
(C-3)
2
3
Ω ( = 125.044 522 22° − 6,962,890.5390" TTT + 7.455TTT
+ 0.008TTT
The nutation parameters are then found using (McCarthy, 1992:33)
a pi = a n1i M ( + a n 2i M O + a n3i µ ( + a n 4i DO + a n5i Ω (
106
∆ψ = ∑ ( A pi + A pli TTDB ) sin( a pi )
i =1
106
∆ε = ∑ ( Aei + Aeli TTDB ) cos(a pi )
(C-4)
i =1
Corrections to the nutation parameters (δ∆ψ1980 and δ∆ε1980) supplied as Earth Orientation Parameters (EOP)
from the IERS are simply added to the resulting values in Eq. 4 to provide compatibility with the newer IAU 2000
Resolutions (Kaplan, 2005). These corrections also include effects from Free Core Nutation (FCN) that correct
errors in the IAU-76 precession and IAU-80 nutation. However for TEME, these corrections do not appear to be
used. The nutation parameters let us find the true obliquity of the ecliptic, ε. (McCarthy, 1992:29–31)
∆ψ 1980 = ∆ψ + δ∆ψ 1980
∆ε 1980 = ∆ε + δ∆ε 1980
3
ε = 84,381.448"−46.8150TTT − 0.000 59TTT2 + 0.001 813TTT
ε = ε + ∆ε 1980
(C-5)
The equation of the equinoxes (EQeqe1980) can then be found. The last two terms in the EQeqe1980 are probably not
included in AFSPC formulations. From McCarthy (1992:30)
EQeqe1980 = ∆ψ 1980 cos(ε ) + 0.002 64" sin(Ω ( ) + 0.000 063 sin( 2Ω ( )
−6 3
2
θ GMST 1982 = 67,310.54841 s + (876,600 h + 8,640,184.812 866 s )TUT 1 + 0.093 104TUT
1 − 6.2 x10 TUT 1
θ GAST 1982 = θ GMST 1982 + EQeqe1980
These relations let us form the transformation equations.
32
(C-6)
[P] MOD − J 2000 = ROT 3(ζ ) ROT 2(−Θ) ROT 3( z )
[N]TOD − MOD = ROT 1(−ε ) ROT 3(∆Ψ ) ROT 1(ε )
K
K
rJ2000 = [P ][N ] ROT 3(− EQeqe1980 ) rTEME
K
K
v J 2000 = [P ][N ] [ ROT 3(− EQeqe1980 )] v TEME
[
]
(C-7)
An example is useful to show the various options and their effect on the resulting vectors. Consider an initial
ECI (J2000.0, IAU76/FK5) state vector.
June 28, 2000, 15: 8:51.655 000 UTC
∆UT1 = 0.162 360s, ∆AT 21s, xp = 0.098 700", yp = 0.286 000"
rJ2000 = 3961.744 260 3 6010.215 610 9 4619.362 575 8 km
vJ2000 = −5.314 643 386 3.964 357 585 1.752 939 153 km/s
Converting to standard TOD, PEF via IAU 76/FK5, without the nutation corrections (δ∆ψ1980 and δ∆ε1980), but using
the two additional terms with EQeqe1980,
JDUT1 = 2,451,724.131 155 293 40, TTT = 0.004 904 360 547
rTOD = 3961.421 498 5 6010.475 268 8 4619.301 531 0 km
vTOD = −5.314 833 569 3.964 181 915 1.752 759 802 km/s
rPEF = 298.803 632 8 −7192.314 622 9 4619.301 531 0 km
vPEF = 6.105 014 271 −0.131 824 177 1.752 759 802 km/s
For the TEME transformation, use equations (3) to (6) to find the approximate parameters (with 4 nutation terms
in Eq (4), no additional two terms in Eq (6), and no small angle approximations). Then, transform TOD or PEF to
TEME using Eq (7) or Eq (2) respectively.
rTEME = 3961.003 549 8 6010.751 174 0 4619.300 930 1 km
vTEME = −5.315 109 069 3.963 813 071 1.752 758 562 km/s
Notice this vector is “in between TOD and PEF, but much closer to the TOD value – a reason it is sometimes
[imprecisely] considered “inertial”. We consider these numbers are within a few mm of CMOC results.
The related issue for TEME ‘of date’ and TEME ‘of epoch’ can also be demonstrated with the following TLE
data at epoch and at 3 days into the future.
1 00005U 58002B
00179.78495062 .00000023 00000-0 28098-4 0 4753
2 00005 34.2682 348.7242 1859667 331.7664 19.3264 10.82419157413667
Some users have assumed comments in CMOC-produced computer outputs suggested TEME of epoch
(precession is used), but most users appear to assume “of date.” These headers often state: “Ephemeris generated
by SGP4 using the WGS-72 earth model. Coordinate frame = true equator and mean equinox of epoch using the
FK5 mean of J2000 time and reference frame.” We believe this statement still exists today. There is no mention of
TEME in the FK5 theory and applicable documents. Analytical Graphic Inc.’s STK provides options for each with
the ‘of date’ option as the default, and we concur with the ‘of date’ position. Although the change between the two
over a week or so is small, it is something measurable if agreement to a centimeter or less is desired. Note that this
ignores the general accuracy of the TLE data being no better than a few kilometers. Using the TLE example data, we
find the TEME vector at a time 3 days in the future (day = 182.784 950 62) from the TLE epoch.
rTEME = −9060.473 735 69 4658.709 525 02 813.686 731 53 km
vTEME = −2.232 832 783 −4.110 453 490 −3.157 345 433 km/s
Next, find the nutation transformation matrix at this time.
[R ]TEME
⎡ 0.999 999 999 56 0.000 000 000 00 0.000 029 505 95⎤
= ⎢⎢− 0.000 000 000 65 0.999 999 999 76 0.000 022 007 05⎥⎥
⎢⎣ − 0.000 029 505 95 0.000 022 007 05 0.999 999 999 32 ⎥⎦
Using Eq (7), we find the inertial (“J2000”) vector at the future time.
rJ2000 = −9059.941 378 6 4659.697 200 0 813.958 887 5 km
vJ2000 = −2.233 348 094 −4.110 136 162 −3.157 394 074 km/s
For the future time using the ’of epoch’ option, the nutation matrix is found at the epoch time as
33
(C-8)
[R ]TEME
⎡ 0.999 999 999 55 0.000 000 000 00 0.000 030 111 90 ⎤
= ⎢⎢ − 0.000 000 000 66 0.999 999 999 76 0.000 021 766 37⎥⎥
⎢⎣− 0.000 030 111 90 0.000 021 766 37 0.999 999 999 31⎥⎦
and the resulting vector is
rJ2000 = −9059.951 079 9 4659.680 755 6 813.945 045 1 km
vJ2000 = −2.233 336 111 −4.110 141 024 −3.157 396 220 km/s
This is a difference of 23.6 m in just 3 days.
34
(C-9)
Appendix D – Test Case Listing
SGP4-VER.TLE –
Test cases include those used for the figures in the paper (with a keyword “## fig”), and those used for
verification to exercise various aspects of the code. The additional values on the second line were added to simplify
automatic processing of each test – the ephemeris starting minutes from epoch (MFE) to the ending MFE, and the
delta time step in minutes.
#
#
1
2
#
1
2
#
#
1
2
#
1
2
#
1
2
#
1
2
#
1
2
#
1
2
#
#
1
2
#
1
2
#
1
2
#
1
2
#
1
2
#
#
1
2
#
1
2
#
1
2
#
1
2
#
#
1
2
#
#
1
2
#
1
2
#
#
1
------------------ Verification test cases ---------------------# TEME example
00005U 58002B
00179.78495062 .00000023 00000-0 28098-4 0 4753
00005 34.2682 348.7242 1859667 331.7664 19.3264 10.82419157413667
## fig show lyddane fix error with gsfc ver
04632U 70093B
04031.91070959 -.00000084 00000-0 10000-3 0 9955
04632 11.4628 273.1101 1450506 207.6000 143.9350 1.20231981 44145
DELTA 1 DEB
# near earth normal drag equation
# perigee = 377.26km, so moderate drag case
06251U 62025E
06176.82412014 .00008885 00000-0 12808-3 0 3985
06251 58.0579 54.0425 0030035 139.1568 221.1854 15.56387291 6774
MOLNIYA 2-14
# 12h resonant ecc in 0.65 to 0.7 range
08195U 75081A
06176.33215444 .00000099 00000-0 11873-3 0
813
08195 64.1586 279.0717 6877146 264.7651 20.2257 2.00491383225656
MOLNIYA 1-36
## fig 12h resonant ecc in 0.7 to 0.715 range
09880U 77021A
06176.56157475 .00000421 00000-0 10000-3 0 9814
09880 64.5968 349.3786 7069051 270.0229 16.3320 2.00813614112380
SMS 1 AKM
# show the integrator problem with gsfc ver
09998U 74033F
05148.79417928 -.00000112 00000-0 00000+0 0 4480
09998
9.4958 313.1750 0270971 327.5225 30.8097 1.16186785 45878
# Original STR#3 SDP4 test
11801U
80230.29629788 .01431103 00000-0 14311-1
13
11801 46.7916 230.4354 7318036 47.4722 10.4117 2.28537848
13
EUTELSAT 1-F1 (ECS1)## fig lyddane choice in GSFC at 2080 min
14128U 83058A
06176.02844893 -.00000158 00000-0 10000-3 0 9627
14128 11.4384 35.2134 0011562 26.4582 333.5652 0.98870114 46093
SL-6 R/B(2)
# Deep space, perigee = 82.48 (<98) for
# s4 > 20 mod
16925U 86065D
06151.67415771 .02550794 -30915-6 18784-3 0 4486
16925 62.0906 295.0239 5596327 245.1593 47.9690 4.88511875148616
SL-12 R/B
# Shows Lyddane choice at 1860 and 4700 min
20413U 83020D
05363.79166667 .00000000 00000-0 00000+0 0 7041
20413 12.3514 187.4253 7864447 196.3027 356.5478 0.24690082 7978
MOLNIYA 1-83
# 12h resonant, ecc > 0.715 (negative BSTAR)
21897U 92011A
06176.02341244 -.00001273 00000-0 -13525-3 0 3044
21897 62.1749 198.0096 7421690 253.0462 20.1561 2.01269994104880
SL-6 R/B(2)
# last tle given, decayed 2006-04-04, day 94
22312U 93002D
06094.46235912 .99999999 81888-5 49949-3 0 3953
22312 62.1486 77.4698 0308723 267.9229 88.7392 15.95744531 98783
SL-6 R/B(2)
# 12h resonant ecc in the > 0.715 range
22674U 93035D
06176.55909107 .00002121 00000-0 29868-3 0 6569
22674 63.5035 354.4452 7541712 253.3264 18.7754 1.96679808 93877
ARIANE 44L+ R/B
# Lyddane bug at <= 70 min for atan2(),
# no quadrant fix
23177U 94040C
06175.45752052 .00000386 00000-0 76590-3 0
95
23177
7.0496 179.8238 7258491 296.0482
8.3061 2.25906668 97438
WIND
# STR#3 Kepler failes past about 200 min
23333U 94071A
94305.49999999 -.00172956 26967-3 10000-3 0
15
23333 28.7490
2.3720 9728298 30.4360
1.3500 0.07309491
70
ARIANE 42P+3 R/B
## fig Lyddane bug at > 280.5 min for AcTan()
23599U 95029B
06171.76535463 .00085586 12891-6 12956-2 0 2905
23599
6.9327
0.2849 5782022 274.4436 25.2425 4.47796565123555
ITALSAT 2
# 24h resonant GEO, inclination > 3 deg
24208U 96044A
06177.04061740 -.00000094 00000-0 10000-3 0 1600
24208
3.8536 80.0121 0026640 311.0977 48.3000 1.00778054 36119
AMC-4
## fig low incl, show incl shift with
## gsfc version from 240 to 1440 min
25954U 99060A
04039.68057285 -.00000108 00000-0 00000-0 0 6847
25954
0.0004 243.8136 0001765 15.5294 22.7134 1.00271289 15615
INTELSAT 902
# negative incl at 9313 min then
# 270 deg Lyddane bug at 37606 min
26900U 01039A
06106.74503247 .00000045 00000-0 10000-3 0 8290
26900
0.0164 266.5378 0003319 86.1794 182.2590 1.00273847 16981
COSMOS 1024 DEB
# 12h resonant ecc in 0.5 to 0.65 range
26975U 78066F
06174.85818871 .00000620 00000-0 10000-3 0 6809
26975 68.4714 236.1303 5602877 123.7484 302.5767 2.05657553 67521
CBERS 2
# Near Earth, ecc = 8.84E-5 (< 1.0e-4)
# drop certain normal drag terms
28057U 03049A
06177.78615833 .00000060 00000-0 35940-4 0 1836
0.00
4320.0
360.00
-5184.0
-4896.0
120.00
0.0
2880.0
120.00
0.0
2880.0
120.00
0.0
2880.0
120.00
-1440.0
-720.00
60.0
0.0
1440.0
360.00
0.0
2880.0
120.00
0.0
1440.0
120.00
1440.0
4320.0
120.00
0.0
2880.0
120.00
54.2028672
1440.0
20.00
0.0
2880.0
120.00
0.0
1440.0
120.00
0.0
1600.0
120.00
0.0
720.0
20.00
0.0
1440.0
120.00
-1440.0
1440.0
120.00
9400.00
60.00
2880.0
120.00
9300.00
0.0
35
2
#
1
2
#
1
2
#
1
2
#
#
1
2
#
#
1
2
#
1
2
#
#
1
2
#
1
2
#
28057 98.4283 247.6961 0000884 88.1964 271.9322 14.35478080140550
NAVSTAR 53 (USA 175)# 12h non-resonant GPS (ecc < 0.5 ecc)
28129U 03058A
06175.57071136 -.00000104 00000-0 10000-3 0
459
28129 54.7298 324.8098 0048506 266.2640 93.1663 2.00562768 18443
COSMOS 2405
# Near Earth, perigee = 127.20 (< 156) s4 mod
28350U 04020A
06167.21788666 .16154492 76267-5 18678-3 0 8894
28350 64.9977 345.6130 0024870 260.7578 99.9590 16.47856722116490
H-2 R/B
# Deep space, perigee = 135.75 (<156) s4 mod
28623U 05006B
06177.81079184 .00637644 69054-6 96390-3 0 6000
28623 28.5200 114.9834 6249053 170.2550 212.8965 3.79477162 12753
XM-3
# 24h resonant geo, incl < 3 deg goes
# negative around 1130 min
28626U 05008A
06176.46683397 -.00000205 00000-0 10000-3 0 2190
28626
0.0019 286.9433 0000335 13.7918 55.6504 1.00270176 4891
MINOTAUR R/B
# Sub-orbital case - Decayed 2005-11-29
#(perigee = -51km), lost in 50 minutes
28872U 05037B
05333.02012661 .25992681 00000-0 24476-3 0 1534
28872 96.4736 157.9986 0303955 244.0492 110.6523 16.46015938 10708
SL-14 DEB
# Last stage of decay - lost in under 420 min
29141U 85108AA 06170.26783845 .99999999 00000-0 13519-0 0
718
29141 82.4288 273.4882 0015848 277.2124 83.9133 15.93343074 6828
SL-12 DEB
# Near Earth, perigee = 212.24 < 220
# simplified drag eq
29238U 06022G
06177.28732010 .00766286 10823-4 13334-2 0
101
29238 51.5595 213.7903 0202579 95.2503 267.9010 15.73823839 1061
# Original STR#3 SGP4 test
88888U
80275.98708465 .00073094 13844-3 66816-4 0
87
88888 72.8435 115.9689 0086731 52.6988 110.5714 16.05824518 1058
0.0
2880.0
120.00
0.0
1440.0
120.00
0.0
2880.0
120.00
0.0
1440.0
120.00
0.0
1440.0
120.00
0.0
50.0
5.00
0.0
440.0
20.00
0.0
1440.0
120.00
0.0
1440.0
120.00
36
Appendix E – Test Case Results Listing
TCPPVER.OUT –
The results are given below for the verification TLE data in Appendix D. Note that this version includes the results of the Lyddane choice using the perturbed
inclination as indicated in the GSFC code, and also uses the WGS-72 constants. There is also a “true” time reference at the end of each line to facilitate the
conversion with minutes from epoch and UTC.
Min from epoch
position x km
position y km
position z km
7022.46529266
-7154.03120202
-7134.59340119
5568.53901181
-938.55923943
-9680.56121728
190.19796988
5579.55640116
-8650.73082219
-5429.79204164
6759.04583722
-3791.44531559
-9060.47373569
-1400.08296755
-3783.17682504
6531.68641334
4492.06992591
-6268.18748831
2802.47771354
7746.96653614
-3995.61396789
-1914.93811525
7574.36493792
2001.58198220
-5712.95617894
4658.70952502
0.03995155
-3536.19412294
3260.27186483
3863.87641983
-4294.02924751
124.10688038
5110.00675412
-1518.82108966
-3007.03603443
3747.39305236
2783.55192533
-4533.48630714
813.68673153
2334.11450085
-29020.02587128
-32982.56870101
-22097.68730513
-15129.94694545
-41920.44035349
13819.84419063
-11125.54996609
-31583.13829284
-36907.74526221
3988.31022699
-3935.69800083
-1675.12766915
4993.62642836
-1115.07959514
-4329.10008198
3692.60030028
2301.83510037
-4990.91637950
642.27769977
4719.78335752
-3299.16993602
-2777.14682335
4992.31573893
-8.22384755
-4966.20137963
2954.49390331
3363.28794321
-4856.66780070
-497.84480071
5241.61936096
-2451.38045953
-3791.87520638
4730.53958356
1159.27802897
5498.96657235
409.10980837
-5683.30432352
2890.54969900
4015.11691491
-5176.70287935
-976.24265255
5723.92394553
-2303.42547880
-4332.89821901
4798.06938996
1576.83168320
-5663.16031708
1716.62356770
4662.21521668
-4379.59155037
-2080.65984650
5559.55841180
-1107.03450192
-4863.46005312
3910.75960683
2610.60463261
-5378.82851382
524.05006433
5056.60175495
vel km/s
vel km/s
vel km/s
year
mon day hr min sec
4.534807250
-2.093935425
-2.557327851
2.744852980
0.989878080
-3.227347517
-0.139152358
4.276837355
-2.515322836
-2.229392830
3.644723952
-0.082384354
-3.157345433
2000
2000
2000
2000
2000
2000
2000
2000
2000
2000
2000
2000
6
6
6
6
6
6
6
6
6
6
6
6
28
28
28
28
29
29
29
29
30
30
30
30
-0.03867437 2.826321032 -0.065091664 0.570936053
-5713.33679183 -1.768068390 -3.235371192 -0.395206135
-6803.28472771 0.617446996 -3.379240041 0.085954707
-4836.34329328 2.230597499 -2.166594667 0.426443070
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2004
2004
2004
2004
1
1
1
1
28 7:27:25.308584
28 9:27:25.308597
28 11:27:25.308611
28 12:15:25.308600
0.90055879
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2006
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2006
2006
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5 xx
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37
0:50:19.733571
6:50:19.733571
12:50:19.733571
18:50:19.733571
0:50:19.733571
6:50:19.733571
12:50:19.733571
18:50:19.733571
0:50:19.733571
6:50:19.733571
12:50:19.733571
18:50:19.733571
21:46:43.980124
23:46:43.980097
1:46:43.980111
3:46:43.980124
5:46:43.980097
7:46:43.980111
9:46:43.980124
11:46:43.980097
13:46:43.980111
15:46:43.980124
17:46:43.980097
19:46:43.980111
21:46:43.980124
23:46:43.980097
1:46:43.980111
3:46:43.980124
5:46:43.980097
7:46:43.980111
9:46:43.980124
11:46:43.980097
13:46:43.980111
15:46:43.980124
17:46:43.980097
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2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
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6
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6
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6
6
6
6
6
6
6
6
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6
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6
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9:58:18.143649
11:58:18.143622
13:58:18.143636
15:58:18.143649
17:58:18.143622
19:58:18.143636
21:58:18.143649
23:58:18.143622
1:58:18.143636
3:58:18.143649
5:58:18.143622
7:58:18.143636
9:58:18.143649
11:58:18.143622
13:58:18.143636
15:58:18.143649
17:58:18.143622
19:58:18.143636
21:58:18.143649
23:58:18.143622
1:58:18.143636
3:58:18.143649
5:58:18.143622
7:58:18.143636
13020.06750784
19190.32482476
11332.67806218
328.74217398
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13725.09398980
19089.63879226
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2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
25
25
25
25
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27
15:28:40.058423
17:28:40.058396
19:28:40.058410
21:28:40.058423
23:28:40.058396
1:28:40.058410
3:28:40.058423
5:28:40.058396
7:28:40.058410
9:28:40.058423
11:28:40.058396
13:28:40.058410
15:28:40.058423
17:28:40.058396
19:28:40.058410
21:28:40.058423
23:28:40.058396
1:28:40.058410
3:28:40.058423
5:28:40.058396
7:28:40.058410
9:28:40.058423
11:28:40.058396
13:28:40.058410
25532.98947267
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309.25349929
11949.04009077
22400.45329336
30640.84752458
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1.903464941 2.634294312
0.545888526
0.267510059
0.402111628
0.497284100
0.542792913
0.534540934
2005
2005
2005
2005
2005
5
5
5
5
5
27
27
27
27
27
19:
20:
21:
22:
23:
38
3:37.089777
3:37.089763
3:37.089790
3:37.089777
3:37.089763
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23341.26015320
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13568.39733054
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2628.58762420
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11801 xx
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9787.87836256
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19433.57330019
0.997393045 3.079858548
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2005
2005
2005
2005
2005
2005
2005
2005
5
5
5
5
5
5
5
5
28
28
28
28
28
28
28
28
5.107155391 6.444680305 -0.186133297
-1.301137319 -1.151315600 -0.283335823
-0.320504528 2.679841539 -2.084054355
-1.016674392 -2.290267981 0.728923337
-1.094251553 0.923589906 -1.522311008
1980
1980
1980
1980
8
8
8
8
17 13: 6:40.136822
17 19: 6:40.136822
18 1: 6:40.136822
18 7: 6:40.136822
-1.32832986
4186.18304085
7273.03412906
8448.49867693
7406.25540271
4422.91711801
282.09554383
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-8447.62393401
-7564.83463543
-4699.87621174
-601.47121821
3654.91125886
6950.29891751
8418.91984963
7677.19769359
4922.96388841
879.86971348
-3393.15365183
-6780.02161760
-8395.46892032
-7815.04516935
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31
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6
6
6
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25
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2:33:42.834827
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4
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12: 0: 0.000000
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4
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16:20:
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20:52:
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-3.405238557 0.888214503
6.493677331 -1.885545282
0.710067769 0.940691824
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
6
6
6
6
6
6
6
6
6
6
6
26
27
27
27
27
27
27
27
27
27
27
23:27:32.414949
1:27:32.414963
3:27:32.414976
5:27:32.414949
7:27:32.414963
9:27:32.414976
11:27:32.414949
13:27:32.414963
15:27:32.414976
17:27:32.414949
19:27:32.414963
42080.71852213
37740.00085593
23232.82515008
2467.44290178
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-35285.00095313
-42103.20138132
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19282.77774728
35480.60990600
42119.96263499
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18802.76872802
35187.33981802
42093.60909959
37661.66243819
23085.44402778
2291.06228893
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0.81851294
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0.000103486
0.000168300
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2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
6
6
6
6
6
6
6
6
6
6
6
6
25
25
25
25
25
25
26
26
26
26
26
26
13:12:14.455025
15:12:14.454998
17:12:14.455012
19:12:14.455025
21:12:14.454998
23:12:14.455012
1:12:14.455025
3:12:14.454998
5:12:14.455012
7:12:14.455025
9:12:14.454998
11:12:14.455012
-6131.82730456
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896.73799533
2896.99663534
4545.78970167
5627.43299371
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2446.52815528
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2420.46580562
1965.98738086
1281.54541294
447.12357305
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-253.64211033 -0.144920228 0.995100963 7.658645067
2011.54515100 2.325207364 -0.047125672 7.296234071
4035.30855837 4.464585796 -1.060923209 6.070907874
5582.12569607 6.049639376 -1.935777558 4.148607019
6474.68172772 6.920746273 -2.580517337 1.748783868
6607.22400507 6.983396282 -2.925846168 -0.872655207
5954.92675486 6.211488246 -2.926949815 -3.433959806
4580.16512984 4.656984233 -2.568711513 -5.638510954
2634.16714930 2.464141047 -1.873985161 -7.195743032
349.87996209 -0.121276950 -0.911981546 -7.859613894
-1979.24314527 -2.763269534 0.199691915 -7.482796996
2005
2005
2005
2005
2005
2005
2005
2005
2005
2005
11
11
11
11
11
11
11
11
11
11
29
29
29
29
29
29
29
29
29
29
0:33:58.939092
0:38:58.939114
0:43:58.939096
0:48:58.939118
0:53:58.939101
0:58:58.939123
1: 3:58.939105
1: 8:58.939127
1:13:58.939109
1:18:58.939092
423.99295524
931.80883587
-83.44906141
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867.44295097
559.16882013
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406.15811659
916.34911813
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884.59720467
446.40767236
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584.40295819
779.59211765
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6286.20208453
3259.26005348
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3376.30587937
6196.00229484
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6304.31821405
2872.17248379
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4086.66839620
5588.35473301
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1100.73728301
6399.18000837
136.13040356
6529.19244527
2223.49837161
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4256.73810533
5699.22017391
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148.33021477
6491.09810362
1960.08739882
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4725.83632696
5093.05596650
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1781.00536019
6311.59529480
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2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
6:45:41.242102
7: 5:41.242111
7:25:41.242079
7:45:41.242088
8: 5:41.242097
8:25:41.242106
8:45:41.242075
9: 5:41.242084
9:25:41.242093
9:45:41.242102
10: 5:41.242111
10:25:41.242079
10:45:41.242088
11: 5:41.242097
11:25:41.242106
11:45:41.242075
12: 5:41.242084
12:25:41.242093
12:45:41.242102
13: 5:41.242111
1.006373613
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1.077709303
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45
420.00000000
29238 xx
0.00000000
120.00000000
240.00000000
360.00000000
480.00000000
600.00000000
720.00000000
840.00000000
960.00000000
1080.00000000
1200.00000000
1320.00000000
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88888 xx
0.00000000
120.00000000
240.00000000
360.00000000
480.00000000
600.00000000
720.00000000
840.00000000
960.00000000
1080.00000000
1200.00000000
1320.00000000
1440.00000000
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192.65232023
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0.396006194 -7.882964919 -0.289331517
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4036.26455287
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6149.65800134
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2328.96975262
1020.69234558
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556.05661780
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328.54999674
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3503.70977525
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3121.73026235
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4532.80979343
1222.89768554
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713.96374435
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196.40072866
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6.044752775
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0.652415093
5.769341172
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1.211072678
2006
6 19 13:25:41.242079
6.023253926
-2.701606741
-3.896001154
5.604262034
-0.350943511
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4.403125405
2.015642495
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2.642104494
4.021854210
-5.842951828
0.577253064
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
2006
6
6
6
6
6
6
6
6
6
6
6
6
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26
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26
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27
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27
27
8:53:44.456634
10:53:44.456607
12:53:44.456620
14:53:44.456634
16:53:44.456607
18:53:44.456620
20:53:44.456634
22:53:44.456607
0:53:44.456620
2:53:44.456634
4:53:44.456607
6:53:44.456620
-7.090816210
1.827985678
5.162585826
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2.397897864
4.700576353
-7.319959258
2.957941672
4.202420227
-7.362824152
3.504058731
3.671022712
-7.356193131
1980
1980
1980
1980
1980
1980
1980
1980
1980
1980
1980
1980
10
10
10
10
10
10
10
10
10
10
10
10
2
2
2
2
2
2
2
2
2
2
2
2
1:41:24.113771
3:41:24.113744
5:41:24.113757
7:41:24.113771
9:41:24.113744
11:41:24.113757
13:41:24.113771
15:41:24.113744
17:41:24.113757
19:41:24.113771
21:41:24.11374
23:41:24.113757
46
Appendix F – Computer Code Listing
Producing computer code in multiple languages is advantageous for testing as many smaller issues were
corrected in this process. Although some features do not exist in each language, an attempt was made to separate the
mathematical theory, the Input/Output, the debugging, and the extra routines for the main program. The debugging
portion is not listed here to reduce the size of the paper, but the full files are available on the website. In addition, the
extra routines (sgp4ext) are not included in this listing as they are primarily intended for use with a main program,
and they vary widely by language. At a future time, it may be advisable to standardize debugging and warning
output routines to handle these cases for integrated programs.
The dependence of each routine is shown below, with parentheses for the name of the file where the routine is
found. This was discussed graphically with Fig. 4 in the paper, and is shown in Fig 12.
SGP4EXT- Misc routines for the main program, math, time, etc. (varies by language for intrinsic math routines)
MAG
CROSS
DOT
ANGLE
NEWTONNU
RV2COE
JDAY
DAYS2MDHMS
INVJDAY
SGP4UNIT- SGP4 mathematical routines including GST and getting the constants.
DPPER, DSCOM, DSPACE, GSTIME, and GETGRAVCONST have no coupling.
DSINIT
– GETGRAVCONST,
INITL
– GETGRAVCONST
– GSTIME (sgp4ext)
SGP4INIT
– GETGRAVCONST
– INITL
– DSCOM
– DPPER
– DSINIT
– SGP4
SGP4
– GETGRAVCONST
– DSPACE
– DPPER
SGP4IO- TLE data parser
TWOLINE2RV
– SGP4INIT (sgp4unit)
– DAYS2MDHMS (sgp4ext)
– JDAY (sgp4ext)
TESTCPP- Main driver for test program (last three letters indicate the language)
MAIN
– TWOLINE2RV (sgp4io)
– SGP4 (sgp4unit)
– INVJDAY (sgp4ext)
– RV2COE (sgp4ext)
47
START
Loop to
read
input file
of TLE
data
TwoLine2RVSGP
4
SGP4init
INITL
Loop
GETGRAVCONST
GSTIME
Days2DMYHMS
if
DSCOM
if
DPPER
if
DSINIT
JDay
Loop to
propagate
each tle
Loop
SGP4
GETGRAVCONST
GETGRAVCONST
SGP4
if
DSPACE
if
DPPER
GETGRAVCONST
DSPACE
DPPER
Function
Locations
SGP4Ext
Output
SGP4IO
SGP4Unit
Figure 12. Program Code Structure. An example flowchart shows the relations between the
various routines in the revised code. Note that the initialization is required a single time after each
new TLE is processed.
48
/*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
--------------------------------------------------------------------testcpp.cpp
this program tests the sgp4 propagator.
companion code for
fundamentals of astrodynamics and applications
2004
by david vallado
(w) 719-573-2600, email [email protected]
*****************************************************************
current :
14 aug 06 david vallado
update mfe for verification time steps, constants
changes :
20 jul 05 david vallado
fixes for paper, corrections from paul crawford
fix record file and get working
14 may 01 david vallado
2nd edition baseline
97 nasa
internet version
80 norad
original baseline
----------------------------------------------------------------
#include
#include
#include
#include
#include
*/
<stdio.h>
<math.h>
<string.h>
<stdlib.h>
<io.h>
#include "sgp4ext.h"
#include "sgp4unit.h"
#include "sgp4io.h"
#define pi 3.14159265358979323846
int main()
{
char str[1];
char infilename[12];
double ro[3];
double vo[3];
char typerun;
gravconsttype whichconst;
int whichcon;
FILE *infile, *outfile, *outfilee;
// ---------------------------- locals ------------------------------double p, a, ecc, incl, node, argp, nu, m, arglat, truelon, lonper;
double sec, jd, rad, tsince, startmfe, stopmfe, deltamin;
int i; int year; int mon; int day; int hr; int min;
char longstr1[130];
typedef char str3[4];
str3 monstr[13];
char outname[64];
char longstr2[130];
elsetrec satrec;
rad = 180.0 / pi;
// -----------------------strcpy(monstr[1], "Jan");
strcpy(monstr[2], "Feb");
strcpy(monstr[3], "Mar");
strcpy(monstr[4], "Apr");
strcpy(monstr[5], "May");
strcpy(monstr[6], "Jun");
strcpy(monstr[7], "Jul");
strcpy(monstr[8], "Aug");
strcpy(monstr[9], "Sep");
strcpy(monstr[10], "Oct");
strcpy(monstr[11], "Nov");
strcpy(monstr[12], "Dec");
implementation
--------------------------
//typerun = 'c' compare 1 year of full satcat data
//typerun = 'v' verification run, requires modified elm file with
//
start stop and delta times
49
printf("input type of run c, v \n");
scanf( "%c",&typerun );
printf("input which constants 72 84 \n");
scanf( "%i",&whichcon );
if (whichcon == 721) whichconst = wgs72old;
if (whichcon == 72) whichconst = wgs72;
if (whichcon == 84) whichconst = wgs84;
// ---------------- setup files for operation -----------------// input 2-line element set file
printf("input elset filename: \n");
scanf( "%s",&infilename );
infile = fopen(infilename, "r");
if (infile == NULL)
{
printf("Failed to open file: %s\n", infilename);
return 1;
}
if (typerun == 'c')
outfile = fopen("tcppall.out", "w");
else
{
if (typerun == 'v')
outfile = fopen("tcppver.out", "w");
else
outfile = fopen("tcpp.out", "w");
}
//
//
dbgfile = fopen("sgp4test.dbg", "w");
fprintf(dbgfile,"this is the debug output\n\n" );
// ----------------- test simple propagation ------------------while (feof(infile) == 0)
{
do
{
fgets( longstr1,130,infile);
strncpy(str, &longstr1[0], 1);
str[1] = '\0';
} while ((strcmp(str, "#")==0)&&(feof(infile) == 0));
if (feof(infile) == 0)
{
fgets( longstr2,130,infile);
// convert the char string to sgp4 elements
// includes initialization of sgp4
twoline2rv( longstr1, longstr2, typerun, whichconst,
startmfe, stopmfe, deltamin, satrec );
fprintf(outfile, "%ld xx\n", satrec.satnum);
printf(" %ld\n", satrec.satnum);
// call the propagator to get the initial state vector value
sgp4 (whichconst, satrec, 0.0, ro, vo);
// generate .e files
jd = satrec.jdsatepoch;
strncpy(outname,&longstr1[2],5);
outname[5]= '.';
outname[6]= 'e';
outname[7]= '\0';
invjday( jd, year,mon,day,hr,min, sec );
outfilee = fopen(outname, "w");
fprintf(outfilee,"stk.v.4.3 \n"); // must use 4.3...
fprintf(outfilee,"\n");
fprintf(outfilee,"BEGIN Ephemeris \n");
fprintf(outfilee," \n");
fprintf(outfilee,"NumberOfEphemerisPoints
146 \n");
fprintf(outfilee,"ScenarioEpoch
%3i %3s%5i%3i:%2i:%12.9f \n",day,monstr[mon],
year,hr,min,sec );
fprintf(outfilee,"InterpolationMethod
Lagrange \n");
fprintf(outfilee,"InterpolationOrder
5 \n");
fprintf(outfilee,"CentralBody
Earth \n");
fprintf(outfilee,"CoordinateSystem
J2000 \n");
fprintf(outfilee,"CoordinateSystemEpoch %3i %3s%5i%3i:%2i:%12.9f \n",day,
monstr[mon],year,hr,min,sec );
fprintf(outfilee,"DistanceUnit
Kilometers \n");
fprintf(outfilee,"EphemerisTimePosVel \n");
fprintf(outfilee, " %16.8f %16.8f %16.8f %16.8f %12.9f %12.9f %12.9f\n",
50
satrec.t,ro[0],ro[1],ro[2],vo[0],vo[1],vo[2]);
fprintf(outfile, " %16.8f %16.8f %16.8f %16.8f %12.9f %12.9f %12.9f\n",
satrec.t,ro[0],ro[1],ro[2],vo[0],vo[1],vo[2]);
tsince = startmfe;
// check so the first value isn't written twice
if ( fabs(tsince) > 1.0e-8 )
tsince = tsince - deltamin;
// loop to perform the propagation
while ((tsince < stopmfe) && (satrec.error == 0))
{
tsince = tsince + deltamin;
if(tsince > stopmfe)
tsince = stopmfe;
sgp4 (whichconst, satrec,
tsince, ro,
vo);
if (satrec.error > 0)
printf("# *** error: t:= %f *** code = %3d\n",
satrec.t, satrec.error);
if (satrec.error == 0)
{
if ((typerun != 'v') && (typerun != 'c'))
{
jd = satrec.jdsatepoch + tsince/1440.0;
fprintf(outfile,
"%5i%3i%3i %2i:%2i:%9.6f %16.8f%16.8f%16.8%12.9f%12.9f%12.9f\n",
year,mon,day,hr,min,sec );
fprintf(outfile, " %16.8f %16.8f %16.8f %16.8f %12.9f %12.9f %12.9f\n",
tsince,ro[0],ro[1],ro[2],vo[0],vo[1],vo[2]);
//
//
}
else
{
jd = satrec.jdsatepoch + tsince/1440.0;
fprintf(outfilee, " %16.6f %16.8f %16.8f %16.8f %12.9f %12.9f %12.9f \n",
tsince*60.0,ro[0],ro[1],ro[2],vo[0],vo[1],vo[2]);
fprintf(outfile, " %16.8f %16.8f %16.8f %16.8f %12.9f %12.9f %12.9f",
tsince,ro[0],ro[1],ro[2],vo[0],vo[1],vo[2]);
rv2coe(ro, vo, p, a, ecc, incl, node, argp, nu, m, arglat, truelon, lonper );
fprintf(outfile, " %14.6f %8.6f %10.5f %10.5f %10.5f %10.5f %10.5f %5i%3i%3i
%2i:%2i:%9.6f\n",
a, ecc, incl*rad, node*rad, argp*rad, nu*rad,
m*rad,year,mon,day,hr,min,sec);
}
} // if satrec.error == 0
} // while propagating the orbit
} // if not eof
fprintf(outfilee," END Ephemeris \n");
fclose (outfilee);
} // while through the input file
}
return 0;
// end testcpp
51
#ifndef _sgp4ext_
#define _sgp4ext_
/*
---------------------------------------------------------------*
*
sgp4ext.h
*
*
this file contains extra routines needed for the main test program for sgp4.
*
q test these routines are derived from the astro libraries.
*
*
companion code for
*
*
2004
*
by david vallado
*
*
*
*
current :
*
*
separate from ast libraries
*
changes :
*
*
original baseline
*
---------------------------------------------------------------*/
#include <string.h>
#include <math.h>
// ------------------------- function decarations ------------------------double
double
void
double
double
void
void
void
void
void
sgn
(
double
);
mag
(
double[]
);
cross
(
double[], double[], double[]
);
dot
(
double[] , double[]
);
angle
(
double[],
double[]
);
newtonnu
(
double ecc, double nu,
double& e0, double& m
);
rv2coe
(
double[], double[],
double&, double&, double&, double&, double&, double&,
double&, double&, double&, double&, double&
);
jday
(
int, int, int, int, int, double, double&
);
days2mdhms
(
int, double, int&, int&, int&, int&, double&
);
invjday
(
double, int&, int&, int&, int&, int&, double&
);
#endif
52
#ifndef _sgp4io_
#define _sgp4io_
/*
---------------------------------------------------------------*
*
sgp4io.h;
*
*
this file contains miscallaneous functions to read two line element
*
sets. while not formerly part of the sgp4 mathematical theory, they are
*
required for practical implemenation.
*
*
*
separate functions, misc doc
*
changes :
*
15 dec 05 david vallado
*
original baseline
*
----------------------------------------------------------------
*/
#include <math.h>
// for several misc routines
// for sgp4init and getgravconst
// ------------------------- function decarations ------------------------void twoline2rv
(
char[130], char[130],
char,
gravconsttype,
double&,
double&,
double&,
elsetrec&
);
#endif
53
/*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
---------------------------------------------------------------sgp4io.cpp
this file contains miscallaneous functions to read two line element
sets. while not formerly part of the sgp4 mathematical theory, they are
required for practical implemenation.
companion code for
2004
by david vallado
current :
14 aug 06
david vallado
separate functions, misc doc
changes :
15 dec 05
david vallado
original baseline
----------------------------------------------------------------
#include "sgp4io.h"
*/
// for several misc routines
// for sgp4init and getgravconst
#define pi 3.14159265358979323846
/* ----------------------------------------------------------------------------*
*
function twoline2rv
*
* this function converts the two line element set character string data to
*
variables and initializes the sgp4 variables. several intermediate varaibles
*
and quantities are determined. note that the result is a structure so multiple
*
satellites can be processed simaltaneously without having to reinitialize. the
*
verification mode is an important option that permits quick checks of any
*
changes to the underlying technical theory. this option works using a
*
modified tle file in which the start, stop, and delta time values are
*
included at the end of the second line of data. this only works with the
*
verification mode. the catalog mode simply propagates from -1440 to 1440 min
*
from epoch and is useful when performing entire catalog runs.
*
* author
: david vallado
719-573-2600
1 mar 2001
*
* inputs
:
*
longstr1
- first line of the tle
*
longstr2
- second line of the tle
*
typerun
- type of run
verification 'v', catalog 'c', 'n'
*
whichconst - which set of constants to use 72, 84
*
* outputs
:
*
satrec
- structure containing all the sgp4 satellite information
*
* coupling
:
*
getgravconst*
days2mdhms - conversion of days to month, day, hour, minute, second
*
jday
- convert day month year hour minute second into julian date
*
sgp4init
- initialize the sgp4 variables
*
* references
:
*
norad spacetrack report #3
*
vallado, crawford, hujsak, kelso 2006
--------------------------------------------------------------------------- */
void twoline2rv
(
char
longstr1[130], char longstr2[130],
char
typerun,
gravconsttype
whichconst,
double& startmfe, double& stopmfe, double& deltamin,
elsetrec& satrec
)
{
const double rad
=
180.0 / pi;
// 57.29577951308230
const double xpdotp = 1440.0 / (2.0 *pi); // 229.1831180523293
double sec, radiusearthkm, tumin, xke, j2, j3, j4, j3oj2;
int cardnumb, numb, j;
long revnum = 0, elnum = 0;
54
char classification, intldesg[11];
int year = 0;
int mon, day, hr, minute, nexp, ibexp;
getgravconst( whichconst, tumin, radiusearthkm, xke, j2, j3, j4, j3oj2 );
satrec.error = 0;
// set the implied decimal points since doing a formated read
// fixes for bad input data values (missing, ...)
for (j = 10; j <= 15; j++)
if (longstr1[j] == ' ')
longstr1[j] = '_';
if (longstr1[44] != ' ')
longstr1[43] = longstr1[44];
longstr1[44] = '.';
if (longstr1[7] == ' ')
longstr1[7] = 'U';
if (longstr1[9] == ' ')
longstr1[9] = '.';
for (j = 45; j <= 49; j++)
longstr1[j] = '0';
if (longstr1[51] == ' ')
longstr1[51] = '0';
if (longstr1[53] != ' ')
longstr1[52] = longstr1[53];
longstr1[53] = '.';
longstr2[25] = '.';
for (j = 26; j <= 32; j++)
longstr2[j] = '0';
if (longstr1[62] == ' ')
longstr1[62] = '0';
if (longstr1[68] == ' ')
longstr1[68] = '0';
sscanf(longstr1,"%2d %5ld %1c %10s %2d %12lf %11lf %7lf %2d %7lf %2d %2d %6ld ",
&cardnumb,&satrec.satnum,&classification, intldesg, &satrec.epochyr,
&satrec.epochdays,&satrec.ndot, &satrec.nddot, &nexp, &satrec.bstar,
&ibexp, &numb, &elnum );
if (typerun == 'v') // run for specified times from the file
{
if (longstr2[52] == ' ')
sscanf(longstr2,"%2d %5ld %9lf %9lf %8lf %9lf %9lf %10lf %6ld %lf %lf %lf \n",
&cardnumb,&satrec.satnum, &satrec.inclo,
&satrec.nodeo,&satrec.ecco, &satrec.argpo, &satrec.mo, &satrec.no,
&revnum, &startmfe, &stopmfe, &deltamin );
else
sscanf(longstr2,"%2d %5ld %9lf %9lf %8lf %9lf %9lf %11lf %6ld %lf %lf %lf \n",
& revnum, &startmfe, &stopmfe, &deltamin );
}
else // simply run -1 day to +1 day or user input times
{
if (longstr2[52] == ' ')
sscanf(longstr2,"%2d %5ld %9lf %9lf %8lf %9lf %9lf %10lf %6ld \n",
&revnum );
else
sscanf(longstr2,"%2d %5ld %9lf %9lf %8lf %9lf %9lf %11lf %6ld \n",
&revnum );
}
// ---- find no, ndot, nddot ---satrec.no
= satrec.no / xpdotp; //* rad/min
satrec.nddot= satrec.nddot * pow(10.0, nexp);
satrec.bstar= satrec.bstar * pow(10.0, ibexp);
// ---- convert to sgp4 units ---satrec.a
= pow( satrec.no*tumin , (-2.0/3.0) );
satrec.ndot = satrec.ndot / (xpdotp*1440.0); //* ? * minperday
satrec.nddot= satrec.nddot / (xpdotp*1440.0*1440);
// ---- find standard orbital elements ----
55
satrec.inclo
satrec.nodeo
satrec.argpo
satrec.mo
=
=
=
=
satrec.inclo / rad;
satrec.nodeo / rad;
satrec.argpo / rad;
satrec.mo
/ rad;
satrec.alta = satrec.a*(1.0 + satrec.ecco*satrec.ecco) - 1.0;
satrec.altp = satrec.a*(1.0 - satrec.ecco*satrec.ecco) - 1.0;
//
//
//
//
//
---------------------------------------------------------------find sgp4epoch time of element set
remember that sgp4 uses units of days from 0 jan 1950 (sgp4epoch)
and minutes from the epoch (time)
----------------------------------------------------------------
// ---- input start stop times manually
if ((typerun != 'v') && (typerun != 'c'))
{
printf("input start min from epoch \n");
scanf( "%lf",&startmfe );
printf("input stop min from epoch \n");
scanf( "%lf",&stopmfe );
printf("input time step in minutes \n");
scanf( "%lf",&deltamin );
}
// ---- perform complete catalog evaluation
if (typerun == 'c')
{
startmfe = -1440.0;
stopmfe = 1440.0;
deltamin =
20.0;
}
// ---------------- temp
// --------- correct fix
if (satrec.epochyr < 50)
year= satrec.epochyr
else
year= satrec.epochyr
fix for years from 1950-2049 ------------------will occur when year is 4-digit in tle --------+ 2000;
+ 1900;
days2mdhms ( year,satrec.epochdays, mon,day,hr,minute,sec );
jday( year,mon,day,hr,minute,sec, satrec.jdsatepoch );
// ---------------- initialize the orbit at sgp4epoch ------------------sgp4init( whichconst, satrec.satnum, satrec.jdsatepoch-2433281.5, satrec.bstar,
satrec.ecco, satrec.argpo, satrec.inclo, satrec.mo, satrec.no,
satrec.nodeo, satrec);
} // end twoline2rv
56
#ifndef _sgp4unit_
#define _sgp4unit_
/*
---------------------------------------------------------------*
*
sgp4unit.h
*
*
this file contains the sgp4 procedures. the code was originally
*
released in the 1980 and 1986 papers. in 1997 and 1998, the updated and
*
copmbined code (sgp4 and sdp4) was released by nasa on the internet.
*
seawifs.gsfc.nasa.gov/~seawifsp/src/bobdays/
*
*
current :
*
*
chg lyddane choice back to strn3, constants, fmod,
*
separate debug and writes, misc doc
*
changes :
*
*
fixes for paper
*
note that each fix is preceded by a
*
comment with "sgp4fix" and an explanation of
*
what was changed
*
*
*
97 nasa
*
internet version
*
80 norad
*
original baseline
*
---------------------------------------------------------------*/
#include <math.h>
#include <stdio.h>
// -------------------------- structure decarations ---------------------------typedef enum
{
wgs72old,
wgs72,
wgs84
} gravconsttype;
typedef struct elsetrec
{
long int satnum;
int
epochyr, epochtynumrev;
int
error;
char
init, method;
/* Near Earth */
int
isimp;
double aycof , con41
delmo , eta
t4cof , t5cof
nodecf;
/* Deep Space
int
irez;
double d2201
d5421
dnodt
plo
si2
xgh3
xl4
double a
bstar
no;
} elsetrec;
, cc1
, cc4
, argpdot, omgcof
, x1mth2 , x7thm1
, cc5
, d2
, d3
, d4
,
, sinmao , t
, t2cof, t3cof ,
, mdot
, nodedot, xlcof , xmcof ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
*/
,
,
,
,
,
,
,
d2211
d5433
domdt
se2
si3
xgh4
xlamo
, altp
, rcse
d3210
dedt
e3
se3
sl2
xh2
zmol
, alta
, inclo
,
,
,
,
,
,
,
d3222
del1
ee2
sgh2
sl3
xh3
zmos
d4410
del2
peo
sgh3
sl4
xi2
atime
,
,
,
,
,
,
,
d4422
del3
pgho
sgh4
gsto
xi3
xli
, epochdays, jdsatepoch
, nodeo
, ecco
,
,
,
,
,
,
,
d5220
didt
pho
sh2
xfact
xl2
xni;
,
,
,
,
,
,
d5232
dmdt
pinco
sh3
xgh2
xl3
, nddot , ndot
, argpo , mo
,
,
,
,
,
,
,
,
// --------------------------- function decarations ---------------------------int sgp4init
(
gravconsttype whichconst,
const int satn,
const double epoch,
const double xbstar, const double xecco, const double xargpo,
const double xinclo, const double xmo,
const double xno,
const double xnodeo,
elsetrec& satrec
);
int sgp4
(
57
elsetrec& satrec, double tsince,
double r[], double v[]
);
double
gstime
(
double
);
void getgravconst
(
gravconsttype,
double&,
double&,
double&,
double&,
double&,
double&,
double&
);
#endif
58
/*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
---------------------------------------------------------------sgp4unit.cpp
this file contains the sgp4 procedures. the code was originally
released in the 1980 and 1986 papers. in 1997 and 1998, the updated and
copmbined code (sgp4 and sdp4) was released by nasa on the internet.
seawifs.gsfc.nasa.gov/~seawifsp/src/bobdays/
companion code for
2004
by david vallado
current :
14 aug 06
david vallado
chg lyddane choice back to strn3, constants, fmod,
separate debug and writes, misc doc
changes :
26 jul 05
david vallado
fixes for paper
note that each fix is preceded by a
comment with "sgp4fix" and an explanation of
what was changed
97 nasa
internet version
80 norad
original baseline
----------------------------------------------------------------
*/
const char help = 'n';
FILE *dbgfile;
#define pi 3.14159265358979323846
/* ----------- local functions - only ever used internally by sgp4 ---------- */
static void dpper
(
double e3,
double ee2,
double peo,
double pgho,
double pho,
double pinco, double plo,
double se2,
double se3,
double sgh2,
double sgh3,
double sgh4,
double sh2,
double sh3,
double si2,
double si3,
double sl2,
double sl3,
double sl4,
double t,
double xgh2,
double xgh3,
double xgh4,
double xh2,
double xh3,
double xi2,
double xi3,
double xl2,
double xl3,
double xl4,
double zmol,
double zmos,
double inclo,
char init,
double& ep,
double& inclp, double& nodep, double& argpp, double& mp
);
static void dscom
(
double epoch, double ep,
double nodep, double np,
double& snodm, double& cnodm,
double& cosomm,double& day,
double& emsq, double& gam,
double& pinco, double& plo,
double& sgh2, double& sgh3,
double& si2,
double& si3,
double& s1,
double& s2,
double& s6,
double& s7,
double& ss4,
double& ss5,
double& sz2,
double& sz3,
double& sz21, double& sz22,
double& sz33, double& xgh2,
double& xh3,
double& xi2,
double& xl4,
double& nm,
double& z11,
double& z12,
double& z23,
double& z31,
double& zmos
);
double argpp,
double tc,
double inclp,
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
sinim,
e3,
peo,
rtemsq,
sgh4,
sl2,
s3,
ss1,
ss6,
sz11,
sz23,
xgh3,
xi3,
z1,
z13,
z32,
cosim,
ee2,
pgho,
se2,
sh2,
sl3,
s4,
ss2,
ss7,
sz12,
sz31,
xgh4,
xl2,
z2,
z21,
z33,
sinomm,
em,
pho,
se3,
sh3,
sl4,
s5,
ss3,
sz1,
sz13,
sz32,
xh2,
xl3,
z3,
z22,
zmol,
static void dsinit
(
59
double cosim, double emsq,
double s3,
double s4,
double ss2,
double ss3,
double sz3,
double sz11,
double sz31,
double sz33,
double mo,
double mdot,
double xpidot, double z1,
double z21,
double z23,
double eccsq, double& em,
double& nm,
double& nodem,
int& irez,
double& atime, double& d2201,
double& d4410, double& d4422,
double& d5433, double& dedt,
double& dnodt, double& domdt,
double& xfact, double& xlamo,
);
static void dspace
(
int irez,
double d2201,
double d4422,
double dedt,
double dmdt,
double t,
double no,
double& atime,
double& mm,
);
double
double
double
double
double
d2211,
d5220,
del1,
dnodt,
tc,
double& em,
double& xni,
double argpo,
double s5,
double ss4,
double sz13,
double t,
double no,
double z3,
double z31,
double& argpm,
double s1,
double sinim,
double ss5,
double sz21,
double tc,
double nodeo,
double z11,
double z33,
double& inclm,
double s2,
double ss1,
double sz1,
double sz23,
double gsto,
double nodedot,
double z13,
double ecco,
double& mm,
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double
double
double
double
double
d2211,
d5220,
didt,
del1,
xli,
d3210,
d5232,
del2,
domdt,
gsto,
double& argpm,
double& nodem,
static void initl
(
int satn,
double ecco,
double epoch, double inclo,
char& method,
double& ainv, double& ao,
double& con41,
double& cosio2,double& eccsq, double& omeosq,
double& rp,
double& rteosq,double& sinio ,
);
double
double
double
double
double
d3210,
d5232,
dmdt,
del2,
xni
d3222,
d5421,
del3,
argpo,
xfact,
double
double
double
double
double
d3222,
d5421,
dndt,
del3,
d4410,
d5433,
didt,
argpdot,
xlamo,
double& inclm, double& xli,
double& dndt, double& nm
double& no,
double& con42, double& cosio,
double& posq,
double& gsto
60
/* ----------------------------------------------------------------------------*
*
procedure dpper
*
* this procedure provides deep space long period periodic contributions
*
to the mean elements. by design, these periodics are zero at epoch.
*
this used to be dscom which included initialization, but it's really a
*
recurring function.
*
* author
: david vallado
719-573-2600
28 jun 2005
*
* inputs
:
*
e3
*
ee2
*
peo
*
pgho
*
pho
*
pinco
*
plo
*
se2 , se3 , sgh2, sgh3, sgh4, sh2, sh3, si2, si3, sl2, sl3, sl4 *
t
*
xh2, xh3, xi2, xi3, xl2, xl3, xl4 *
zmol
*
zmos
*
ep
- eccentricity
0.0 - 1.0
*
inclo
- inclination - needed for lyddane modification
*
nodep
- right ascension of ascending node
*
argpp
- argument of perigee
*
mp
- mean anomaly
*
* outputs
:
*
ep
- eccentricity
0.0 - 1.0
*
inclp
- inclination
*
nodep
*
argpp
*
mp
- mean anomaly
*
* locals
:
*
alfdp
*
betdp
*
cosip , sinip , cosop , sinop ,
*
dalf
*
dbet
*
dls
*
f2, f3
*
pe
*
pgh
*
ph
*
pinc
*
pl
*
sel
, ses
, sghl , sghs , shl
, shs
, sil
, sinzf , sis
,
*
sll
, sls
*
xls
*
xnoh
*
zf
*
zm
*
* coupling
:
*
none.
*
* references
:
*
hoots, roehrich, norad spacetrack report #3 1980
*
hoots, norad spacetrack report #6 1986
*
hoots, schumacher and glover 2004
*
----------------------------------------------------------------------------*/
static void dpper
(
double e3,
double ee2,
double peo,
double pgho,
double pho,
double pinco, double plo,
double se2,
double se3,
double sgh2,
double sgh3,
double sgh4,
double sh2,
double sh3,
double si2,
double si3,
double sl2,
double sl3,
double sl4,
double t,
double xgh2,
double xgh3,
double xgh4,
double xh2,
double xh3,
double xi2,
double xi3,
double xl2,
double xl3,
double xl4,
double zmol,
double zmos,
double inclo,
char init,
double& ep,
double& inclp, double& nodep, double& argpp, double& mp
)
{
/* --------------------- local variables ------------------------ */
61
const double twopi = 2.0 * pi;
char ildm;
double alfdp, betdp, cosip, cosop, dalf, dbet, dls,
f2,
f3,
pe,
pgh,
ph,
pinc, pl ,
sel,
ses,
sghl, sghs, shll, shs, sil,
sinip, sinop, sinzf, sis,
sll, sls, xls,
xnoh, zf,
zm,
zel,
zes, znl, zns;
/* ---------------------- constants ----------------------------- */
zns
= 1.19459e-5;
zes
= 0.01675;
znl
= 1.5835218e-4;
zel
= 0.05490;
/* --------------- calculate time varying periodics ----------- */
zm
= zmos + zns * t;
// be sure that the initial call has time set to zero
if (init == 'y')
zm = zmos;
zf
= zm + 2.0 * zes * sin(zm);
sinzf = sin(zf);
f2
= 0.5 * sinzf * sinzf - 0.25;
f3
= -0.5 * sinzf * cos(zf);
ses
= se2* f2 + se3 * f3;
sis
= si2 * f2 + si3 * f3;
sls
= sl2 * f2 + sl3 * f3 + sl4 * sinzf;
sghs = sgh2 * f2 + sgh3 * f3 + sgh4 * sinzf;
shs
= sh2 * f2 + sh3 * f3;
zm
= zmol + znl * t;
if (init == 'y')
zm = zmol;
zf
= zm + 2.0 * zel * sin(zm);
sinzf = sin(zf);
f2
= 0.5 * sinzf * sinzf - 0.25;
f3
= -0.5 * sinzf * cos(zf);
sel
= ee2 * f2 + e3 * f3;
sil
= xi2 * f2 + xi3 * f3;
sll
= xl2 * f2 + xl3 * f3 + xl4 * sinzf;
sghl = xgh2 * f2 + xgh3 * f3 + xgh4 * sinzf;
shll = xh2 * f2 + xh3 * f3;
pe
= ses + sel;
pinc = sis + sil;
pl
= sls + sll;
pgh
= sghs + sghl;
ph
= shs + shll;
if (init == 'n')
{
// 0.2 rad = 11.45916 deg
// sgp4fix for lyddane choice
// add next three lines to set up use of original inclination per strn3 ver
ildm = 'y';
if (inclo >= 0.2)
ildm = 'n';
pe
pinc
pl
pgh
ph
inclp
ep
sinip
cosip
=
=
=
=
=
=
=
=
=
pe - peo;
pinc - pinco;
pl - plo;
pgh - pgho;
ph - pho;
inclp + pinc;
ep + pe;
sin(inclp);
cos(inclp);
/* ----------------- apply periodics directly ------------ */
// sgp4fix for lyddane choice
// strn3 used original inclination - this is technically feasible
// gsfc used perturbed inclination - also technically feasible
// probably best to readjust the 0.2 limit value and limit discontinuity
// use next line for original strn3 approach and original inclination
// if (inclo >= 0.2)
// use next line for gsfc version and perturbed inclination
if (inclp >= 0.2)
{
ph
= ph / sinip;
pgh
= pgh - cosip * ph;
argpp = argpp + pgh;
nodep = nodep + ph;
mp
= mp + pl;
}
62
else
{
/* ---- apply periodics with lyddane modification ---- */
sinop = sin(nodep);
cosop = cos(nodep);
alfdp = sinip * sinop;
betdp = sinip * cosop;
dalf
= ph * cosop + pinc * cosip * sinop;
dbet
= -ph * sinop + pinc * cosip * cosop;
alfdp = alfdp + dalf;
betdp = betdp + dbet;
nodep = fmod(nodep, twopi);
xls
= mp + argpp + cosip * nodep;
dls
= pl + pgh - pinc * nodep * sinip;
xls
= xls + dls;
xnoh
= nodep;
nodep = atan2(alfdp, betdp);
if (fabs(xnoh - nodep) > pi)
if (nodep < xnoh)
nodep = nodep + twopi;
else
nodep = nodep - twopi;
mp
= mp + pl;
argpp = xls - mp - cosip * nodep;
}
}
// if init == 'n'
//#include "debug1.cpp"
} // end dpper
63
/*----------------------------------------------------------------------------*
*
procedure dscom
*
* this procedure provides deep space common items used by both the secular
*
and periodics subroutines. input is provided as shown. this routine
*
used to be called dpper, but the functions inside weren't well organized.
*
* author
: david vallado
719-573-2600
28 jun 2005
*
* inputs
:
*
epoch
*
ep
- eccentricity
*
argpp
*
tc
*
inclp
- inclination
*
nodep
*
np
- mean motion
*
* outputs
:
*
sinim , cosim , sinomm , cosomm , snodm , cnodm
*
day
*
e3
*
ee2
*
em
- eccentricity
*
emsq
- eccentricity squared
*
gam
*
peo
*
pgho
*
pho
*
pinco
*
plo
*
rtemsq
*
se2, se3
*
sgh2, sgh3, sgh4
*
sh2, sh3, si2, si3, sl2, sl3, sl4
*
s1, s2, s3, s4, s5, s6, s7
*
ss1, ss2, ss3, ss4, ss5, ss6, ss7, sz1, sz2, sz3
*
sz11, sz12, sz13, sz21, sz22, sz23, sz31, sz32, sz33
*
xgh2, xgh3, xgh4, xh2, xh3, xi2, xi3, xl2, xl3, xl4
*
nm
- mean motion
*
z1, z2, z3, z11, z12, z13, z21, z22, z23, z31, z32, z33
*
zmol
*
zmos
*
* locals
:
*
a1, a2, a3, a4, a5, a6, a7, a8, a9, a10
*
betasq
*
cc
*
ctem, stem
*
x1, x2, x3, x4, x5, x6, x7, x8
*
xnodce
*
xnoi
*
zcosg , zsing , zcosgl , zsingl , zcosh , zsinh , zcoshl , zsinhl ,
*
zcosi , zsini , zcosil , zsinil ,
*
zx
*
zy
*
* coupling
:
*
none.
*
* references
:
*
*
*
*
----------------------------------------------------------------------------*/
static void dscom
(
double epoch, double ep,
double nodep, double np,
double& snodm, double& cnodm,
double& cosomm,double& day,
double& emsq, double& gam,
double& pinco, double& plo,
double& sgh2, double& sgh3,
double& si2,
double& si3,
double& s1,
double& s2,
double& s6,
double& s7,
double& ss4,
double& ss5,
double argpp,
double tc,
double inclp,
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
sinim,
e3,
peo,
rtemsq,
sgh4,
sl2,
s3,
ss1,
ss6,
cosim,
ee2,
pgho,
se2,
sh2,
sl3,
s4,
ss2,
ss7,
sinomm,
em,
pho,
se3,
sh3,
sl4,
s5,
ss3,
sz1,
64
double&
double&
double&
double&
double&
double&
double&
double&
sz2,
sz21,
sz33,
xh3,
xl4,
z11,
z23,
zmos
double&
double&
double&
double&
double&
double&
double&
sz3,
sz22,
xgh2,
xi2,
nm,
z12,
z31,
double&
double&
double&
double&
double&
double&
double&
sz11,
sz23,
xgh3,
xi3,
z1,
z13,
z32,
double&
double&
double&
double&
double&
double&
double&
sz12,
sz31,
xgh4,
xl2,
z2,
z21,
z33,
double&
double&
double&
double&
double&
double&
double&
sz13,
sz32,
xh2,
xl3,
z3,
z22,
zmol,
)
{
/* -------------------------- constants ------------------------- */
const double zes
= 0.01675;
const double zel
= 0.05490;
const double c1ss
= 2.9864797e-6;
const double c1l
= 4.7968065e-7;
const double zsinis = 0.39785416;
const double zcosis = 0.91744867;
const double zcosgs = 0.1945905;
const double zsings = -0.98088458;
const double twopi
= 2.0 * pi;
/* --------------------- local variables ------------------------ */
int lsflg;
double a1
, a2
, a3
, a4
, a5
, a6
, a7
,
a8
, a9
, a10
, betasq, cc
, ctem , stem ,
x1
, x2
, x3
, x4
, x5
, x6
, x7
,
x8
, xnodce, xnoi , zcosg , zcosgl, zcosh , zcoshl,
zcosi , zcosil, zsing , zsingl, zsinh , zsinhl, zsini ,
zsinil, zx
, zy;
nm
em
snodm
cnodm
sinomm
cosomm
sinim
cosim
emsq
betasq
rtemsq
=
=
=
=
=
=
=
=
=
=
=
np;
ep;
sin(nodep);
cos(nodep);
sin(argpp);
cos(argpp);
sin(inclp);
cos(inclp);
em * em;
1.0 - emsq;
sqrt(betasq);
/* ----------------- initialize lunar solar terms --------------- */
peo
= 0.0;
pinco = 0.0;
plo
= 0.0;
pgho
= 0.0;
pho
= 0.0;
day
= epoch + 18261.5 + tc / 1440.0;
xnodce = fmod(4.5236020 - 9.2422029e-4 * day, twopi);
stem
= sin(xnodce);
ctem
= cos(xnodce);
zcosil = 0.91375164 - 0.03568096 * ctem;
zsinil = sqrt(1.0 - zcosil * zcosil);
zsinhl = 0.089683511 * stem / zsinil;
zcoshl = sqrt(1.0 - zsinhl * zsinhl);
gam
= 5.8351514 + 0.0019443680 * day;
zx
= 0.39785416 * stem / zsinil;
zy
= zcoshl * ctem + 0.91744867 * zsinhl * stem;
zx
= atan2(zx, zy);
zx
= gam + zx - xnodce;
zcosgl = cos(zx);
zsingl = sin(zx);
/* ------------------------- do solar terms --------------------- */
zcosg = zcosgs;
zsing = zsings;
zcosi = zcosis;
zsini = zsinis;
zcosh = cnodm;
zsinh = snodm;
cc
= c1ss;
xnoi = 1.0 / nm;
for (lsflg = 1; lsflg <= 2;
{
a1 =
zcosg * zcosh +
a3 = -zsing * zcosh +
a7 = -zcosg * zsinh +
a8 =
zsing * zsini;
a9 =
zsing * zsinh +
lsflg++)
zsing * zcosi * zsinh;
zcosg * zcosi * zsinh;
zsing * zcosi * zcosh;
zcosg * zcosi * zcosh;
65
a10
a2
a4
a5
a6
=
=
=
=
=
x1
x2
x3
x4
x5
x6
x7
x8
= a1 * cosomm +
= a3 * cosomm +
= -a1 * sinomm +
= -a3 * sinomm +
= a5 * sinomm;
= a6 * sinomm;
= a5 * cosomm;
= a6 * cosomm;
z31
z32
z33
z1
z2
z3
z11
z12
=
=
=
=
=
=
=
=
z13 =
z21 =
z22 =
z23
z1
z2
z3
s3
s2
s4
s1
s5
s6
s7
=
=
=
=
=
=
=
=
=
=
=
zcosg
cosim
cosim
-sinim
-sinim
*
*
*
*
*
zsini;
a7 + sinim
a9 + sinim
a7 + cosim
a9 + cosim
a2
a4
a2
a4
*
*
*
*
*
*
*
*
a8;
a10;
a8;
a10;
sinomm;
sinomm;
cosomm;
cosomm;
12.0 * x1 * x1 - 3.0 * x3 * x3;
24.0 * x1 * x2 - 6.0 * x3 * x4;
12.0 * x2 * x2 - 3.0 * x4 * x4;
3.0 * (a1 * a1 + a2 * a2) + z31 * emsq;
6.0 * (a1 * a3 + a2 * a4) + z32 * emsq;
3.0 * (a3 * a3 + a4 * a4) + z33 * emsq;
-6.0 * a1 * a5 + emsq * (-24.0 * x1 * x7-6.0 * x3 * x5);
-6.0 * (a1 * a6 + a3 * a5) + emsq *
(-24.0 * (x2 * x7 + x1 * x8) - 6.0 * (x3 * x6 + x4 * x5));
-6.0 * a3 * a6 + emsq * (-24.0 * x2 * x8 - 6.0 * x4 * x6);
6.0 * a2 * a5 + emsq * (24.0 * x1 * x5 - 6.0 * x3 * x7);
6.0 * (a4 * a5 + a2 * a6) + emsq *
(24.0 * (x2 * x5 + x1 * x6) - 6.0 * (x4 * x7 + x3 * x8));
6.0 * a4 * a6 + emsq * (24.0 * x2 * x6 - 6.0 * x4 * x8);
z1 + z1 + betasq * z31;
z2 + z2 + betasq * z32;
z3 + z3 + betasq * z33;
cc * xnoi;
-0.5 * s3 / rtemsq;
s3 * rtemsq;
-15.0 * em * s4;
x1 * x3 + x2 * x4;
x2 * x3 + x1 * x4;
x2 * x4 - x1 * x3;
/* ----------------------- do lunar terms ------------------- */
if (lsflg == 1)
{
ss1
= s1;
ss2
= s2;
ss3
= s3;
ss4
= s4;
ss5
= s5;
ss6
= s6;
ss7
= s7;
sz1
= z1;
sz2
= z2;
sz3
= z3;
sz11 = z11;
sz12 = z12;
sz13 = z13;
sz21 = z21;
sz22 = z22;
sz23 = z23;
sz31 = z31;
sz32 = z32;
sz33 = z33;
zcosg = zcosgl;
zsing = zsingl;
zcosi = zcosil;
zsini = zsinil;
zcosh = zcoshl * cnodm + zsinhl * snodm;
zsinh = snodm * zcoshl - cnodm * zsinhl;
cc
= c1l;
}
}
zmol = fmod(4.7199672 + 0.22997150 * day - gam, twopi);
zmos = fmod(6.2565837 + 0.017201977 * day, twopi);
/* ------------------------ do solar terms ---------------------- */
se2 =
2.0 * ss1 * ss6;
se3 =
2.0 * ss1 * ss7;
si2 =
2.0 * ss2 * sz12;
si3 =
2.0 * ss2 * (sz13 - sz11);
sl2 = -2.0 * ss3 * sz2;
66
sl3
sl4
sgh2
sgh3
sgh4
sh2
sh3
= -2.0 * ss3
= -2.0 * ss3
=
2.0 * ss4
=
2.0 * ss4
= -18.0 * ss4
= -2.0 * ss2
= -2.0 * ss2
*
*
*
*
*
*
*
(sz3 - sz1);
(-21.0 - 9.0 * emsq) * zes;
sz32;
(sz33 - sz31);
zes;
sz22;
(sz23 - sz21);
/* ------------------------ do lunar terms ---------------------- */
ee2 =
2.0 * s1 * s6;
e3
=
2.0 * s1 * s7;
xi2 =
2.0 * s2 * z12;
xi3 =
2.0 * s2 * (z13 - z11);
xl2 = -2.0 * s3 * z2;
xl3 = -2.0 * s3 * (z3 - z1);
xl4 = -2.0 * s3 * (-21.0 - 9.0 * emsq) * zel;
xgh2 =
2.0 * s4 * z32;
xgh3 =
2.0 * s4 * (z33 - z31);
xgh4 = -18.0 * s4 * zel;
xh2 = -2.0 * s2 * z22;
xh3 = -2.0 * s2 * (z23 - z21);
} // end dscom
67
/*----------------------------------------------------------------------------*
*
procedure dsinit
*
* this procedure provides deep space contributions to mean motion dot due
*
to geopotential resonance with half day and one day orbits.
*
* author
: david vallado
719-573-2600
28 jun 2005
*
* inputs
:
*
cosim, sinim*
emsq
*
argpo
*
s1, s2, s3, s4, s5
*
ss1, ss2, ss3, ss4, ss5 *
sz1, sz3, sz11, sz13, sz21, sz23, sz31, sz33 *
t
- time
*
tc
*
gsto
- greenwich sidereal time
rad
*
mo
- mean anomaly
*
mdot
- mean anomaly dot (rate)
*
no
- mean motion
*
nodeo
*
nodedot
- right ascension of ascending node dot (rate)
*
xpidot
*
z1, z3, z11, z13, z21, z23, z31, z33 *
eccm
- eccentricity
*
argpm
*
inclm
- inclination
*
mm
- mean anomaly
*
xn
- mean motion
*
nodem
*
* outputs
:
*
em
- eccentricity
*
argpm
*
inclm
- inclination
*
mm
- mean anomaly
*
nm
- mean motion
*
nodem
*
irez
- flag for resonance
0-none, 1-one day, 2-half day
*
atime
*
d2201, d2211, d3210, d3222, d4410, d4422, d5220, d5232, d5421, d5433
*
dedt
*
didt
*
dmdt
*
dndt
*
dnodt
*
domdt
*
del1, del2, del3
*
ses , sghl , sghs , sgs , shl , shs , sis , sls
*
theta
*
xfact
*
xlamo
*
xli
*
xni
*
* locals
:
*
ainv2
*
aonv
*
cosisq
*
eoc
*
f220, f221, f311, f321, f322, f330, f441, f442, f522, f523, f542, f543 *
g200, g201, g211, g300, g310, g322, g410, g422, g520, g521, g532, g533 *
sini2
*
temp
*
temp1
*
theta
*
xno2
*
* coupling
:
*
getgravconst
*
* references
:
*
*
*
*
----------------------------------------------------------------------------*/
static void dsinit
68
(
double cosim, double emsq,
double s3,
double s4,
double ss2,
double ss3,
double sz3,
double sz11,
double sz31,
double sz33,
double mo,
double mdot,
double xpidot, double z1,
double z21,
double z23,
double eccsq, double& em,
double& nm,
double& nodem,
int& irez,
double& atime, double& d2201,
double& d4410, double& d4422,
double& d5433, double& dedt,
double& dnodt, double& domdt,
double& xfact, double& xlamo,
double argpo,
double s5,
double ss4,
double sz13,
double t,
double no,
double z3,
double z31,
double& argpm,
double s1,
double sinim,
double ss5,
double sz21,
double tc,
double nodeo,
double z11,
double z33,
double& inclm,
double s2,
double ss1,
double sz1,
double sz23,
double gsto,
double nodedot,
double z13,
double ecco,
double& mm,
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
double&
d2211,
d5220,
didt,
del1,
xli,
d3210,
d5232,
dmdt,
del2,
xni
d3222,
d5421,
dndt,
del3,
)
{
/* --------------------- local variables ------------------------ */
double ainv2 , aonv=0.0, cosisq, eoc, f220 , f221
f321 , f322 , f330 , f441 , f442 , f522
f542 , f543 , g200 , g201 , g211 , g300
g322 , g410 , g422 , g520 , g521 , g532
ses
, sgs
, sghl , sghs , shs
, shll
sini2 , sls
, temp , temp1 , theta , xno2
q31
, q33
, root22, root44, root54, rptim
root52, x2o3 , xke
, znl
, emo
, zns
tumin, radiusearthkm, j2, j3, j4, j3oj2;
q22
q31
q33
root22
root44
root54
rptim
root32
root52
x2o3
znl
zns
=
=
=
=
=
=
=
=
=
=
=
=
,
,
,
,
,
,
,
,
f311 ,
f523 ,
g310 ,
g533 ,
sis
,
q22
,
root32,
emsqo,
1.7891679e-6;
2.1460748e-6;
2.2123015e-7;
1.7891679e-6;
7.3636953e-9;
2.1765803e-9;
4.37526908801129966e-3; // this equates to 7.29211514668855e-5 rad/sec
3.7393792e-7;
1.1428639e-7;
2.0 / 3.0;
1.5835218e-4;
1.19459e-5;
// sgp4fix identify constants and allow alternate values
/* -------------------- deep space initialization ------------ */
irez = 0;
if ((nm < 0.0052359877) && (nm > 0.0034906585))
irez = 1;
if ((nm >= 8.26e-3) && (nm <= 9.24e-3) && (em >= 0.5))
irez = 2;
/* ------------------------ do solar terms ------------------- */
ses = ss1 * zns * ss5;
sis = ss2 * zns * (sz11 + sz13);
sls = -zns * ss3 * (sz1 + sz3 - 14.0 - 6.0 * emsq);
sghs = ss4 * zns * (sz31 + sz33 - 6.0);
shs = -zns * ss2 * (sz21 + sz23);
// sgp4fix for 180 deg incl
if ((inclm < 5.2359877e-2) || (inclm > pi - 5.2359877e-2))
shs = 0.0;
if (sinim != 0.0)
shs = shs / sinim;
sgs = sghs - cosim * shs;
/* ------------------------- do lunar terms ------------------ */
dedt = ses + s1 * znl * s5;
didt = sis + s2 * znl * (z11 + z13);
dmdt = sls - znl * s3 * (z1 + z3 - 14.0 - 6.0 * emsq);
sghl = s4 * znl * (z31 + z33 - 6.0);
shll = -znl * s2 * (z21 + z23);
// sgp4fix for 180 deg incl
if ((inclm < 5.2359877e-2) || (inclm > pi - 5.2359877e-2))
shll = 0.0;
domdt = sgs + sghl;
dnodt = shs;
if (sinim != 0.0)
69
{
domdt = domdt - cosim / sinim * shll;
dnodt = dnodt + shll / sinim;
}
/* ----------- calculate deep space resonance effects -------- */
dndt
= 0.0;
theta = fmod(gsto + tc * rptim, twopi);
em
= em + dedt * t;
inclm = inclm + didt * t;
argpm = argpm + domdt * t;
nodem = nodem + dnodt * t;
mm
= mm + dmdt * t;
//
sgp4fix for negative inclinations
//
the following if statement should be commented out
//if (inclm < 0.0)
// {
//
inclm = -inclm;
//
argpm = argpm - pi;
//
nodem = nodem + pi;
// }
/* -------------- initialize the resonance terms ------------- */
if (irez != 0)
{
aonv = pow(nm / xke, x2o3);
/* ---------- geopotential resonance for 12 hour orbits ------ */
if (irez == 2)
{
cosisq = cosim * cosim;
emo
= em;
em
= ecco;
emsqo = emsq;
emsq
= eccsq;
eoc
= em * emsq;
g201
= -0.306 - (em - 0.64) * 0.440;
if (em <= 0.65)
{
g211 =
3.616 - 13.2470 * em + 16.2900 * emsq;
g310 = -19.302 + 117.3900 * em - 228.4190 * emsq + 156.5910 * eoc;
g322 = -18.9068 + 109.7927 * em - 214.6334 * emsq + 146.5816 * eoc;
g410 = -41.122 + 242.6940 * em - 471.0940 * emsq + 313.9530 * eoc;
g422 = -146.407 + 841.8800 * em - 1629.014 * emsq + 1083.4350 * eoc;
g520 = -532.114 + 3017.977 * em - 5740.032 * emsq + 3708.2760 * eoc;
}
else
{
g211 =
-72.099 +
331.819 * em 508.738 * emsq +
266.724 * eoc;
g310 = -346.844 + 1582.851 * em - 2415.925 * emsq + 1246.113 * eoc;
g322 = -342.585 + 1554.908 * em - 2366.899 * emsq + 1215.972 * eoc;
g410 = -1052.797 + 4758.686 * em - 7193.992 * emsq + 3651.957 * eoc;
g422 = -3581.690 + 16178.110 * em - 24462.770 * emsq + 12422.520 * eoc;
if (em > 0.715)
g520 =-5149.66 + 29936.92 * em - 54087.36 * emsq + 31324.56 * eoc;
else
g520 = 1464.74 - 4664.75 * em + 3763.64 * emsq;
}
if (em < 0.7)
{
g533 = -919.22770 + 4988.6100 * em - 9064.7700 * emsq + 5542.21 * eoc;
g521 = -822.71072 + 4568.6173 * em - 8491.4146 * emsq + 5337.524 * eoc;
g532 = -853.66600 + 4690.2500 * em - 8624.7700 * emsq + 5341.4 * eoc;
}
else
{
g533 =-37995.780 + 161616.52 * em - 229838.20 * emsq + 109377.94 * eoc;
g521 =-51752.104 + 218913.95 * em - 309468.16 * emsq + 146349.42 * eoc;
g532 =-40023.880 + 170470.89 * em - 242699.48 * emsq + 115605.82 * eoc;
}
sini2= sinim * sinim;
f220 = 0.75 * (1.0 + 2.0 * cosim+cosisq);
f221 = 1.5 * sini2;
f321 = 1.875 * sinim * (1.0 - 2.0 * cosim - 3.0 * cosisq);
f322 = -1.875 * sinim * (1.0 + 2.0 * cosim - 3.0 * cosisq);
f441 = 35.0 * sini2 * f220;
f442 = 39.3750 * sini2 * sini2;
f522 = 9.84375 * sinim * (sini2 * (1.0 - 2.0 * cosim- 5.0 * cosisq) +
0.33333333 * (-2.0 + 4.0 * cosim + 6.0 * cosisq) );
70
f523 = sinim * (4.92187512 * sini2 * (-2.0 - 4.0 * cosim +
10.0 * cosisq) + 6.56250012 * (1.0+2.0 * cosim - 3.0 * cosisq));
f542 = 29.53125 * sinim * (2.0 - 8.0 * cosim+cosisq *
(-12.0 + 8.0 * cosim + 10.0 * cosisq));
f543 = 29.53125 * sinim * (-2.0 - 8.0 * cosim+cosisq *
(12.0 + 8.0 * cosim - 10.0 * cosisq));
xno2 = nm * nm;
ainv2 = aonv * aonv;
temp1 = 3.0 * xno2 * ainv2;
temp = temp1 * root22;
d2201 = temp * f220 * g201;
d2211 = temp * f221 * g211;
temp1 = temp1 * aonv;
d3210 = temp * f321 * g310;
d3222 = temp * f322 * g322;
temp = 2.0 * temp1 * root44;
d4410 = temp * f441 * g410;
d4422 = temp * f442 * g422;
d5220 = temp * f522 * g520;
d5232 = temp * f523 * g532;
temp = 2.0 * temp1 * root54;
d5421 = temp * f542 * g521;
d5433 = temp * f543 * g533;
xlamo = fmod(mo + nodeo + nodeo-theta - theta, twopi);
xfact = mdot + dmdt + 2.0 * (nodedot + dnodt - rptim) - no;
em
= emo;
emsq = emsqo;
}
/* ---------------- synchronous resonance terms -------------- */
if (irez == 1)
{
g200 = 1.0 + emsq * (-2.5 + 0.8125 * emsq);
g310 = 1.0 + 2.0 * emsq;
g300 = 1.0 + emsq * (-6.0 + 6.60937 * emsq);
f220 = 0.75 * (1.0 + cosim) * (1.0 + cosim);
f311 = 0.9375 * sinim * sinim * (1.0 + 3.0 * cosim) - 0.75 * (1.0 + cosim);
f330 = 1.0 + cosim;
f330 = 1.875 * f330 * f330 * f330;
del1 = 3.0 * nm * nm * aonv * aonv;
del2 = 2.0 * del1 * f220 * g200 * q22;
del3 = 3.0 * del1 * f330 * g300 * q33 * aonv;
del1 = del1 * f311 * g310 * q31 * aonv;
xlamo = fmod(mo + nodeo + argpo - theta, twopi);
xfact = mdot + xpidot - rptim + dmdt + domdt + dnodt - no;
}
/* ------------ for sgp4, initialize the integrator ---------- */
xli
= xlamo;
xni
= no;
atime = 0.0;
nm
= no + dndt;
}
} // end dsinit
71
/*----------------------------------------------------------------------------*
*
procedure dspace
*
* this procedure provides deep space contributions to mean elements for
*
perturbing third body. these effects have been averaged over one
*
revolution of the sun and moon. for earth resonance effects, the
*
effects have been averaged over no revolutions of the satellite.
*
(mean motion)
*
* author
: david vallado
719-573-2600
28 jun 2005
*
* inputs
:
*
d2201, d2211, d3210, d3222, d4410, d4422, d5220, d5232, d5421, d5433 *
dedt
*
del1, del2, del3 *
didt
*
dmdt
*
dnodt
*
domdt
*
irez
- flag for resonance
0-none, 1-one day, 2-half day
*
argpo
*
argpdot
- argument of perigee dot (rate)
*
t
- time
*
tc
*
gsto
- gst
*
xfact
*
xlamo
*
no
- mean motion
*
atime
*
em
- eccentricity
*
ft
*
argpm
*
inclm
- inclination
*
xli
*
mm
- mean anomaly
*
xni
- mean motion
*
nodem
*
* outputs
:
*
atime
*
em
- eccentricity
*
argpm
*
inclm
- inclination
*
xli
*
mm
- mean anomaly
*
xni
*
nodem
*
dndt
*
nm
- mean motion
*
* locals
:
*
delt
*
ft
*
theta
*
x2li
*
x2omi
*
xl
*
xldot
*
xnddt
*
xndt
*
xomi
*
* coupling
:
*
none
*
* references
:
*
*
*
*
----------------------------------------------------------------------------*/
static void dspace
(
int irez,
double d2201,
double d4422,
double dedt,
double dmdt,
double t,
double
double
double
double
double
d2211,
d5220,
del1,
dnodt,
tc,
double
double
double
double
double
d3210,
d5232,
del2,
domdt,
gsto,
double
double
double
double
double
d3222,
d5421,
del3,
argpo,
xfact,
double
double
double
double
double
d4410,
d5433,
didt,
argpdot,
xlamo,
72
double no,
double& atime, double& em,
double& mm,
double& xni,
double& argpm,
double& nodem,
double& inclm, double& xli,
double& dndt, double& nm
)
{
int iretn , iret;
double delt, ft, theta, x2li, x2omi, xl, xldot , xnddt, xndt, xomi, g22, g32,
g44, g52, g54, fasx2, fasx4, fasx6, rptim , step2, stepn , stepp;
ft
fasx2
fasx4
fasx6
g22
g32
g44
g52
g54
rptim
stepp
stepn
step2
=
=
=
=
=
=
=
=
=
=
=
=
=
0.0;
0.13130908;
2.8843198;
0.37448087;
5.7686396;
0.95240898;
1.8014998;
1.0508330;
4.4108898;
4.37526908801129966e-3; // this equates to 7.29211514668855e-5 rad/sec
720.0;
-720.0;
259200.0;
/* ----------- calculate deep space resonance effects ----------- */
dndt
= 0.0;
theta = fmod(gsto + tc * rptim, twopi);
em
= em + dedt * t;
inclm
argpm
nodem
mm
=
=
=
=
inclm + didt * t;
argpm + domdt * t;
nodem + dnodt * t;
mm + dmdt * t;
//
sgp4fix for negative inclinations
//
the following if statement should be commented out
// if (inclm < 0.0)
// {
//
inclm = -inclm;
//
argpm = argpm - pi;
//
nodem = nodem + pi;
// }
/* - update resonances : numerical (euler-maclaurin) integration - */
/* ------------------------- epoch restart ---------------------- */
//
sgp4fix for propagator problems
//
the following integration works for negative time steps and periods
//
the specific changes are unknown because the original code was so convoluted
ft
= 0.0;
atime = 0.0;
if (irez != 0)
{
if ((atime == 0.0) || ((t >= 0.0) && (atime < 0.0)) ||
((t < 0.0) && (atime >= 0.0)))
{
if (t >= 0.0)
delt = stepp;
else
delt = stepn;
atime = 0.0;
xni
= no;
xli
= xlamo;
}
iretn = 381; // added for do loop
iret =
0; // added for loop
while (iretn == 381)
{
if ((fabs(t) < fabs(atime)) || (iret == 351))
{
if (t >= 0.0)
delt = stepn;
else
delt = stepp;
iret = 351;
iretn = 381;
}
else
{
if (t > 0.0) // error if prev if has atime:=0.0 and t:=0.0 (ge)
delt = stepp;
73
else
delt = stepn;
if (fabs(t - atime) >= stepp)
{
iret = 0;
iretn = 381;
}
else
{
ft
= t - atime;
iretn = 0;
}
}
/* ------------------- dot terms calculated ------------- */
/* ----------- near - synchronous resonance terms ------- */
if (irez != 2)
{
xndt = del1 * sin(xli - fasx2) + del2 * sin(2.0 * (xli - fasx4)) +
del3 * sin(3.0 * (xli - fasx6));
xldot = xni + xfact;
xnddt = del1 * cos(xli - fasx2) +
2.0 * del2 * cos(2.0 * (xli - fasx4)) +
3.0 * del3 * cos(3.0 * (xli - fasx6));
xnddt = xnddt * xldot;
}
else
{
/* --------- near - half-day resonance terms -------- */
xomi = argpo + argpdot * atime;
x2omi = xomi + xomi;
x2li = xli + xli;
xndt = d2201 * sin(x2omi + xli - g22) + d2211 * sin(xli - g22) +
d3210 * sin(xomi + xli - g32) + d3222 * sin(-xomi + xli - g32)+
d4410 * sin(x2omi + x2li - g44)+ d4422 * sin(x2li - g44) +
d5220 * sin(xomi + xli - g52) + d5232 * sin(-xomi + xli - g52)+
d5421 * sin(xomi + x2li - g54) + d5433 * sin(-xomi + x2li - g54);
xldot = xni + xfact;
xnddt = d2201 * cos(x2omi + xli - g22) + d2211 * cos(xli - g22) +
d3210 * cos(xomi + xli - g32) + d3222 * cos(-xomi + xli - g32) +
d5220 * cos(xomi + xli - g52) + d5232 * cos(-xomi + xli - g52) +
2.0 * (d4410 * cos(x2omi + x2li - g44) +
d4422 * cos(x2li - g44) + d5421 * cos(xomi + x2li - g54) +
d5433 * cos(-xomi + x2li - g54));
xnddt = xnddt * xldot;
}
/* ----------------------- integrator ------------------- */
if (iretn == 381)
{
xli
= xli + xldot * delt + xndt * step2;
xni
= xni + xndt * delt + xnddt * step2;
atime = atime + delt;
}
} // while iretn = 381
nm = xni + xndt * ft + xnddt * ft * ft * 0.5;
xl = xli + xldot * ft + xndt * ft * ft * 0.5;
if (irez != 1)
{
mm
= xl - 2.0 * nodem + 2.0 * theta;
dndt = nm - no;
}
else
{
mm
= xl - nodem - argpm + theta;
dndt = nm - no;
}
nm = no + dndt;
}
} // end dsspace
74
/*----------------------------------------------------------------------------*
*
procedure initl
*
* this procedure initializes the spg4 propagator. all the initialization is
*
consolidated here instead of having multiple loops inside other routines.
*
* author
: david vallado
719-573-2600
28 jun 2005
*
* inputs
:
*
ecco
- eccentricity
0.0 - 1.0
*
epoch
- epoch time in days from jan 0, 1950. 0 hr
*
inclo
- inclination of satellite
*
no
- mean motion of satellite
*
satn
- satellite number
*
* outputs
:
*
ainv
- 1.0 / a
*
ao
- semi major axis
*
con41
*
con42
- 1.0 - 5.0 cos(i)
*
cosio
- cosine of inclination
*
cosio2
- cosio squared
*
eccsq
*
method
- flag for deep space
'd', 'n'
*
omeosq
- 1.0 - ecco * ecco
*
posq
- semi-parameter squared
*
rp
- radius of perigee
*
rteosq
- square root of (1.0 - ecco*ecco)
*
sinio
- sine of inclination
*
gsto
- gst at time of observation
rad
*
no
- mean motion of satellite
*
* locals
:
*
ak
*
d1
*
del
*
adel
*
po
*
* coupling
:
*
getgravconst
*
gstime
- find greenwich sidereal time from the julian date
*
* references
:
*
*
*
*
----------------------------------------------------------------------------*/
static void initl
(
int satn,
double ecco,
double epoch, double inclo,
double& no,
char& method,
double& ainv, double& ao,
double& con41, double& con42, double& cosio,
double& cosio2,double& eccsq, double& omeosq, double& posq,
double& rp,
double& rteosq,double& sinio , double& gsto
)
{
/* --------------------- local variables ------------------------ */
double ak, d1, del, adel, po, x2o3, j2, xke,
tumin, radiusearthkm, j3, j4, j3oj2;
/* ----------------------- earth constants ---------------------- */
x2o3
= 2.0 / 3.0;
/* ------------- calculate auxillary epoch quantities ---------- */
eccsq = ecco * ecco;
omeosq = 1.0 - eccsq;
rteosq = sqrt(omeosq);
cosio = cos(inclo);
cosio2 = cosio * cosio;
/* ------------------ un-kozai the mean motion ----------------- */
ak
= pow(xke / no, x2o3);
d1
= 0.75 * j2 * (3.0 * cosio2 - 1.0) / (rteosq * omeosq);
del
= d1 / (ak * ak);
75
adel
del
no
= ak * (1.0 - del * del - del *
(1.0 / 3.0 + 134.0 * del * del / 81.0));
= d1/(adel * adel);
= no / (1.0 + del);
ao
= pow(xke / no, x2o3);
sinio = sin(inclo);
po
= ao * omeosq;
con42 = 1.0 - 5.0 * cosio2;
con41 = -con42-cosio2-cosio2;
ainv = 1.0 / ao;
posq = po * po;
rp
= ao * (1.0 - ecco);
method = 'n';
gsto = gstime(epoch + 2433281.5);
} // end initl
76
/*----------------------------------------------------------------------------*
*
procedure sgp4init
*
* this procedure initializes variables for sgp4.
*
* author
: david vallado
719-573-2600
28 jun 2005
*
* inputs
:
*
satn
- satellite number
*
bstar
- sgp4 type drag coefficient
kg/m2er
*
ecco
- eccentricity
*
epoch
- epoch time in days from jan 0, 1950. 0 hr
*
argpo
- argument of perigee (output if ds)
*
inclo
- inclination
*
mo
- mean anomaly (output if ds)
*
no
- mean motion
*
nodeo
*
* outputs
:
*
satrec
- common values for subsequent calls
*
return code - non-zero on error.
*
1 - mean elements, ecc >= 1.0 or ecc < -0.001 or a < 0.95 er
*
2 - mean motion less than 0.0
*
3 - pert elements, ecc < 0.0 or ecc > 1.0
*
4 - semi-latus rectum < 0.0
*
5 - epoch elements are sub-orbital
*
6 - satellite has decayed
*
* locals
:
*
cnodm , snodm , cosim , sinim , cosomm , sinomm
*
cc1sq , cc2
, cc3
*
coef
, coef1
*
cosio4
*
day
*
dndt
*
em
- eccentricity
*
emsq
*
eeta
*
etasq
*
gam
*
argpm
*
nodem
*
inclm
- inclination
*
mm
- mean anomaly
*
nm
- mean motion
*
perige
- perigee
*
pinvsq
*
psisq
*
qzms24
*
rtemsq
*
s1, s2, s3, s4, s5, s6, s7
*
sfour
*
ss1, ss2, ss3, ss4, ss5, ss6, ss7
*
sz1, sz2, sz3
*
sz11, sz12, sz13, sz21, sz22, sz23, sz31, sz32, sz33
*
tc
*
temp
*
temp1, temp2, temp3
*
tsi
*
xpidot
*
xhdot1
*
z1, z2, z3
*
z11, z12, z13, z21, z22, z23, z31, z32, z33
*
* coupling
:
*
getgravconst*
initl
*
dscom
*
dpper
*
dsinit
*
sgp4
*
* references
:
*
*
*
*
----------------------------------------------------------------------------*/
int sgp4init
77
(
const int satn,
const double epoch,
const double xbstar, const double xecco, const double xargpo,
const double xinclo, const double xmo,
const double xno,
const double xnodeo, elsetrec& satrec
)
{
/* --------------------- local variables ------------------------ */
double ao, ainv,
con42, cosio, sinio, cosio2, eccsq,
omeosq, posq,
rp,
rteosq,
cnodm , snodm , cosim , sinim , cosomm, sinomm, cc1sq ,
cc2
, cc3
, coef , coef1 , cosio4, day
, dndt ,
em
, emsq , eeta , etasq , gam
, argpm , nodem ,
inclm , mm
, nm
, perige, pinvsq, psisq , qzms24,
rtemsq, s1
, s2
, s3
, s4
, s5
, s6
,
s7
, sfour , ss1
, ss2
, ss3
, ss4
, ss5
,
ss6
, ss7
, sz1
, sz2
, sz3
, sz11 , sz12 ,
sz13 , sz21 , sz22 , sz23 , sz31 , sz32 , sz33 ,
tc
, temp , temp1 , temp2 , temp3 , tsi
, xpidot,
xhdot1, z1
, z2
, z3
, z11
, z12
, z13
,
z21
, z22
, z23
, z31
, z32
, z33,
qzms2t, ss, j2, j3oj2, j4, x2o3, r[3], v[3],
tumin, radiusearthkm, xke, j3;
/* ------------------------ initialization --------------------- */
// sgp4fix divisor for divide by zero check on inclination
const double temp4
=
1.0 + cos(pi-1.0e-9);
/* ----------satrec.isimp
satrec.con41
satrec.cc5
satrec.d4
satrec.argpdot
satrec.t
satrec.t4cof
satrec.x7thm1
satrec.xlcof
set all near earth variables to zero ------------ */
= 0;
satrec.method = 'n'; satrec.aycof
= 0.0;
= 0.0; satrec.cc1
= 0.0; satrec.cc4
= 0.0;
= 0.0; satrec.d2
= 0.0; satrec.d3
= 0.0;
= 0.0; satrec.delmo = 0.0; satrec.eta
= 0.0;
= 0.0; satrec.omgcof = 0.0; satrec.sinmao
= 0.0;
= 0.0; satrec.t2cof = 0.0; satrec.t3cof
= 0.0;
= 0.0; satrec.t5cof = 0.0; satrec.x1mth2
= 0.0;
= 0.0; satrec.mdot
= 0.0; satrec.nodedot = 0.0;
= 0.0; satrec.xmcof = 0.0; satrec.nodecf
= 0.0;
/* ----------satrec.irez =
satrec.d3210 =
satrec.d4422 =
satrec.d5421 =
satrec.del1 =
satrec.didt =
satrec.domdt =
satrec.peo
=
satrec.pinco =
satrec.se3
=
satrec.sgh4 =
satrec.si2
=
satrec.sl3
=
satrec.xfact =
satrec.xgh4 =
satrec.xi2
=
satrec.xl3
=
satrec.zmol =
satrec.xli
=
set all deep space variables to zero ------------ */
0;
satrec.d2201 = 0.0; satrec.d2211 = 0.0;
0.0; satrec.d3222 = 0.0; satrec.d4410 = 0.0;
0.0; satrec.d5220 = 0.0; satrec.d5232 = 0.0;
0.0; satrec.d5433 = 0.0; satrec.dedt = 0.0;
0.0; satrec.del2 = 0.0; satrec.del3 = 0.0;
0.0; satrec.dmdt = 0.0; satrec.dnodt = 0.0;
0.0; satrec.e3
= 0.0; satrec.ee2
= 0.0;
0.0; satrec.pgho = 0.0; satrec.pho
= 0.0;
0.0; satrec.plo
= 0.0; satrec.se2
= 0.0;
0.0; satrec.sgh2 = 0.0; satrec.sgh3 = 0.0;
0.0; satrec.sh2
= 0.0; satrec.sh3
= 0.0;
0.0; satrec.si3
= 0.0; satrec.sl2
= 0.0;
0.0; satrec.sl4
= 0.0; satrec.gsto = 0.0;
0.0; satrec.xgh2 = 0.0; satrec.xgh3 = 0.0;
0.0; satrec.xh2
= 0.0; satrec.xh3
= 0.0;
0.0; satrec.xi3
= 0.0; satrec.xl2
= 0.0;
0.0; satrec.xl4
= 0.0; satrec.xlamo = 0.0;
0.0; satrec.zmos = 0.0; satrec.atime = 0.0;
0.0; satrec.xni
= 0.0;
// sgp4fix - note the following variables are also passed directly via satrec.
// it is possible to streamline the sgp4init call by deleting the "x"
// variables, but the user would need to set the satrec.* values first. we
// include the additional assignments in case twoline2rv is not used.
satrec.bstar
= xbstar;
satrec.ecco
= xecco;
satrec.argpo
= xargpo;
satrec.inclo
= xinclo;
satrec.mo
= xmo;
satrec.no
= xno;
satrec.nodeo
= xnodeo;
/* ------------------------ earth constants ----------------------- */
ss
= 78.0 / radiusearthkm + 1.0;
qzms2t = pow(((120.0 - 78.0) / radiusearthkm), 4);
x2o3
= 2.0 / 3.0;
satrec.init = 'y';
satrec.t = 0.0;
78
initl
(
satn, whichconst, satrec.ecco, epoch, satrec.inclo, satrec.no, satrec.method,
ainv, ao, satrec.con41, con42, cosio, cosio2, eccsq, omeosq,
posq, rp, rteosq, sinio, satrec.gsto
);
satrec.error = 0;
//
if (rp < 1.0)
{
printf("# *** satn%d epoch elts sub-orbital ***\n", satn);
satrec.error = 5;
}
if ((omeosq >= 0.0 ) || ( satrec.no >= 0.0))
{
satrec.isimp = 0;
if (rp < (220.0 / radiusearthkm + 1.0))
satrec.isimp = 1;
sfour = ss;
qzms24 = qzms2t;
perige = (rp - 1.0) * radiusearthkm;
/* - for perigees below 156 km, s and qoms2t are altered - */
if (perige < 156.0)
{
sfour = perige - 78.0;
if (perige < 98.0)
sfour = 20.0;
qzms24 = pow(((120.0 - sfour) / radiusearthkm), 4.0);
sfour = sfour / radiusearthkm + 1.0;
}
pinvsq = 1.0 / posq;
tsi = 1.0 / (ao - sfour);
satrec.eta = ao * satrec.ecco * tsi;
etasq = satrec.eta * satrec.eta;
eeta = satrec.ecco * satrec.eta;
psisq = fabs(1.0 - etasq);
coef = qzms24 * pow(tsi, 4.0);
coef1 = coef / pow(psisq, 3.5);
cc2
= coef1 * satrec.no * (ao * (1.0 + 1.5 * etasq + eeta *
(4.0 + etasq)) + 0.375 * j2 * tsi / psisq * satrec.con41 *
(8.0 + 3.0 * etasq * (8.0 + etasq)));
satrec.cc1
= satrec.bstar * cc2;
cc3
= 0.0;
if (satrec.ecco > 1.0e-4)
cc3 = -2.0 * coef * tsi * j3oj2 * satrec.no * sinio / satrec.ecco;
satrec.x1mth2 = 1.0 - cosio2;
satrec.cc4
= 2.0* satrec.no * coef1 * ao * omeosq *
(satrec.eta * (2.0 + 0.5 * etasq) + satrec.ecco *
(0.5 + 2.0 * etasq) - j2 * tsi / (ao * psisq) *
(-3.0 * satrec.con41 * (1.0 - 2.0 * eeta + etasq *
(1.5 - 0.5 * eeta)) + 0.75 * satrec.x1mth2 *
(2.0 * etasq - eeta * (1.0 + etasq)) * cos(2.0 * satrec.argpo)));
satrec.cc5 = 2.0 * coef1 * ao * omeosq * (1.0 + 2.75 *
(etasq + eeta) + eeta * etasq);
cosio4 = cosio2 * cosio2;
temp1 = 1.5 * j2 * pinvsq * satrec.no;
temp2 = 0.5 * temp1 * j2 * pinvsq;
temp3 = -0.46875 * j4 * pinvsq * pinvsq * satrec.no;
satrec.mdot
= satrec.no + 0.5 * temp1 * rteosq * satrec.con41 + 0.0625 *
temp2 * rteosq * (13.0 - 78.0 * cosio2 + 137.0 * cosio4);
satrec.argpdot = -0.5 * temp1 * con42 + 0.0625 * temp2 *
(7.0 - 114.0 * cosio2 + 395.0 * cosio4) +
temp3 * (3.0 - 36.0 * cosio2 + 49.0 * cosio4);
xhdot1
= -temp1 * cosio;
satrec.nodedot = xhdot1 + (0.5 * temp2 * (4.0 - 19.0 * cosio2) +
2.0 * temp3 * (3.0 - 7.0 * cosio2)) * cosio;
xpidot
= satrec.argpdot+ satrec.nodedot;
satrec.omgcof
= satrec.bstar * cc3 * cos(satrec.argpo);
satrec.xmcof
= 0.0;
if (satrec.ecco > 1.0e-4)
satrec.xmcof = -x2o3 * coef * satrec.bstar / eeta;
satrec.nodecf = 3.5 * omeosq * xhdot1 * satrec.cc1;
satrec.t2cof
= 1.5 * satrec.cc1;
// sgp4fix for divide by zero with xinco = 180 deg
if (fabs(cosio+1.0) > 1.5e-12)
satrec.xlcof = -0.25 * j3oj2 * sinio * (3.0 + 5.0 * cosio) / (1.0 + cosio);
else
79
satrec.xlcof
satrec.aycof
=
satrec.delmo
=
satrec.sinmao =
satrec.x7thm1 =
= -0.25 * j3oj2 * sinio * (3.0 + 5.0 * cosio) / temp4;
-0.5 * j3oj2 * sinio;
pow((1.0 + satrec.eta * cos(satrec.mo)), 3);
sin(satrec.mo);
7.0 * cosio2 - 1.0;
/* --------------- deep space initialization ------------- */
if ((2*pi / satrec.no) >= 225.0)
{
satrec.method = 'd';
satrec.isimp = 1;
tc
= 0.0;
inclm = satrec.inclo;
dscom
(
epoch, satrec.ecco, satrec.argpo, tc, satrec.inclo, satrec.nodeo,
satrec.no, snodm, cnodm, sinim, cosim,sinomm,
cosomm,
day, satrec.e3, satrec.ee2, em,
emsq, gam,
satrec.peo, satrec.pgho,
satrec.pho, satrec.pinco,
satrec.plo, rtemsq,
satrec.se2, satrec.se3,
satrec.sgh2, satrec.sgh3,
satrec.sgh4,
satrec.sh2, satrec.sh3,
satrec.si2, satrec.si3,
satrec.sl2, satrec.sl3,
satrec.sl4, s1, s2, s3, s4, s5,
s6,
s7,
ss1, ss2, ss3, ss4, ss5, ss6, ss7, sz1, sz2, sz3,
sz11, sz12, sz13, sz21, sz22, sz23, sz31, sz32, sz33,
satrec.xgh2, satrec.xgh3,
satrec.xgh4, satrec.xh2,
satrec.xh3, satrec.xi2,
satrec.xi3, satrec.xl2,
satrec.xl3, satrec.xl4,
nm, z1, z2, z3, z11,
z12, z13, z21, z22, z23, z31, z32, z33,
satrec.zmol, satrec.zmos
);
dpper
(
satrec.e3, satrec.ee2, satrec.peo, satrec.pgho,
satrec.pho, satrec.pinco, satrec.plo, satrec.se2,
satrec.se3, satrec.sgh2, satrec.sgh3, satrec.sgh4,
satrec.sh2, satrec.sh3, satrec.si2, satrec.si3,
satrec.sl2, satrec.sl3, satrec.sl4, satrec.t,
satrec.xgh2,satrec.xgh3,satrec.xgh4, satrec.xh2,
satrec.xh3, satrec.xi2, satrec.xi3, satrec.xl2,
satrec.xl3, satrec.xl4, satrec.zmol, satrec.zmos, inclm, satrec.init,
satrec.ecco, satrec.inclo, satrec.nodeo, satrec.argpo, satrec.mo
);
argpm
nodem
mm
= 0.0;
= 0.0;
= 0.0;
dsinit
(
whichconst,
cosim, emsq, satrec.argpo, s1, s2, s3, s4, s5, sinim, ss1, ss2, ss3, ss4,
ss5, sz1, sz3, sz11, sz13, sz21, sz23, sz31, sz33, satrec.t, tc,
satrec.gsto, satrec.mo, satrec.mdot, satrec.no, satrec.nodeo,
satrec.nodedot, xpidot, z1, z3, z11, z13, z21, z23, z31, z33,
satrec.ecco, eccsq, em, argpm, inclm, mm, nm, nodem,
satrec.irez, satrec.atime,
satrec.d2201, satrec.d2211, satrec.d3210, satrec.d3222 ,
satrec.d4410, satrec.d4422, satrec.d5220, satrec.d5232,
satrec.d5421, satrec.d5433, satrec.dedt, satrec.didt,
satrec.dmdt, dndt,
satrec.dnodt, satrec.domdt ,
satrec.del1, satrec.del2, satrec.del3, satrec.xfact,
satrec.xlamo, satrec.xli,
satrec.xni
);
}
/* ----------- set variables if not deep space ----------- */
if (satrec.isimp != 1)
{
cc1sq
= satrec.cc1 * satrec.cc1;
satrec.d2
= 4.0 * ao * tsi * cc1sq;
temp
= satrec.d2 * tsi * satrec.cc1 / 3.0;
satrec.d3
= (17.0 * ao + sfour) * temp;
satrec.d4
= 0.5 * temp * ao * tsi * (221.0 * ao + 31.0 * sfour) *
satrec.cc1;
satrec.t3cof = satrec.d2 + 2.0 * cc1sq;
satrec.t4cof = 0.25 * (3.0 * satrec.d3 + satrec.cc1 *
(12.0 * satrec.d2 + 10.0 * cc1sq));
satrec.t5cof = 0.2 * (3.0 * satrec.d4 +
12.0 * satrec.cc1 * satrec.d3 +
80
6.0 * satrec.d2 * satrec.d2 +
15.0 * cc1sq * (2.0 * satrec.d2 + cc1sq));
}
} // if omeosq = 0 ...
/* finally propogate to zero epoch to initialise all others. */
if(satrec.error == 0)
sgp4(whichconst, satrec, 0.0, r, v);
satrec.init = 'n';
return satrec.error;
} // end sgp4init
81
/*----------------------------------------------------------------------------*
*
procedure sgp4
*
* this procedure is the sgp4 prediction model from space command. this is an
*
updated and combined version of sgp4 and sdp4, which were originally
*
published separately in spacetrack report #3. this version follows the nasa
*
release on the internet. there are a few fixes that are added to correct
*
known errors in the existing implementations.
*
* author
: david vallado
719-573-2600
28 jun 2005
*
* inputs
:
*
satrec
- initialised structure from sgp4init() call.
*
tsince
- time eince epoch (minutes)
*
* outputs
:
*
r
- position vector
km
*
v
- velocity
km/sec
* return code - non-zero on error.
*
1 - mean elements, ecc >= 1.0 or ecc < -0.001 or a < 0.95 er
*
2 - mean motion less than 0.0
*
3 - pert elements, ecc < 0.0 or ecc > 1.0
*
4 - semi-latus rectum < 0.0
*
5 - epoch elements are sub-orbital
*
6 - satellite has decayed
*
* locals
:
*
am
*
axnl, aynl
*
betal
*
cosim
, sinim
, cosomm , sinomm , cnod
, snod
, cos2u
,
*
sin2u
, coseo1 , sineo1 , cosi
, sini
, cosip
, sinip
,
*
cosisq , cossu
, sinsu
, cosu
, sinu
*
delm
*
delomg
*
dndt
*
eccm
*
emsq
*
ecose
*
el2
*
eo1
*
eccp
*
esine
*
argpm
*
argpp
*
omgadf
*
pl
*
r
*
rtemsq
*
rdotl
*
rl
*
rvdot
*
rvdotl
*
su
*
t2 , t3
, t4
, tc
*
tem5, temp , temp1 , temp2 , tempa , tempe , templ
*
u
, ux
, uy
, uz
, vx
, vy
, vz
*
inclm
- inclination
*
mm
- mean anomaly
*
nm
- mean motion
*
nodem
- right asc of ascending node
*
xinc
*
xincp
*
xl
*
xlm
*
mp
*
xmdf
*
xmx
*
xmy
*
nodedf
*
xnode
*
nodep
*
np
*
* coupling
:
*
getgravconst*
dpper
*
dpspace
*
* references
:
82
*
*
*
*
----------------------------------------------------------------------------*/
int sgp4
(
gravconsttype whichconst, elsetrec& satrec, double tsince,
double r[3], double v[3]
)
{
double am
, axnl , aynl , betal , cosim , cnod ,
cos2u, coseo1, cosi , cosip , cosisq, cossu , cosu,
delm , delomg, em
, emsq , ecose , el2
, eo1 ,
ep
, esine , argpm, argpp , argpdf, pl,
mrt = 0.0,
mvt , rdotl , rl
, rvdot , rvdotl, sinim ,
sin2u, sineo1, sini , sinip , sinsu , sinu ,
snod , su
, t2
, t3
, t4
, tem5 , temp,
temp1, temp2 , tempa, tempe , templ , u
, ux ,
uy
, uz
, vx
, vy
, vz
, inclm , mm ,
nm
, nodem, xinc , xincp , xl
, xlm
, mp ,
xmdf , xmx
, xmy , nodedf, xnode , nodep, tc , dndt,
twopi, x2o3 , j2
, j3
, tumin, j4 , xke
, j3oj2, radiusearthkm,
vkmpersec;
int ktr;
/* ------------------ set mathematical constants --------------- */
// sgp4fix divisor for divide by zero check on inclination
const double temp4
=
1.0 + cos(pi-1.0e-9);
twopi = 2.0 * pi;
x2o3 = 2.0 / 3.0;
vkmpersec
= radiusearthkm * xke/60.0;
/* --------------------- clear sgp4 error flag ----------------- */
satrec.t
= tsince;
satrec.error = 0;
/* ------- update for secular gravity and atmospheric drag ----- */
xmdf
= satrec.mo + satrec.mdot * satrec.t;
argpdf = satrec.argpo + satrec.argpdot * satrec.t;
nodedf = satrec.nodeo + satrec.nodedot * satrec.t;
argpm
= argpdf;
mm
= xmdf;
t2
= satrec.t * satrec.t;
nodem
= nodedf + satrec.nodecf * t2;
tempa
= 1.0 - satrec.cc1 * satrec.t;
tempe
= satrec.bstar * satrec.cc4 * satrec.t;
templ
= satrec.t2cof * t2;
if (satrec.isimp != 1)
{
delomg = satrec.omgcof * satrec.t;
delm
= satrec.xmcof *
(pow((1.0 + satrec.eta * cos(xmdf)), 3) satrec.delmo);
temp
= delomg + delm;
mm
= xmdf + temp;
argpm = argpdf - temp;
t3
= t2 * satrec.t;
t4
= t3 * satrec.t;
tempa = tempa - satrec.d2 * t2 - satrec.d3 * t3 satrec.d4 * t4;
tempe = tempe + satrec.bstar * satrec.cc5 * (sin(mm) satrec.sinmao);
templ = templ + satrec.t3cof * t3 + t4 * (satrec.t4cof +
satrec.t * satrec.t5cof);
}
nm
= satrec.no;
em
= satrec.ecco;
inclm = satrec.inclo;
if (satrec.method == 'd')
{
tc = satrec.t;
dspace
(
satrec.irez,
satrec.d2201, satrec.d2211, satrec.d3210,
83
satrec.d5433, satrec.dedt, satrec.del1,
satrec.del2, satrec.del3, satrec.didt,
satrec.dmdt, satrec.dnodt, satrec.domdt,
satrec.argpo, satrec.argpdot, satrec.t, tc,
satrec.gsto, satrec.xfact, satrec.xlamo,
satrec.no, satrec.atime,
em, argpm, inclm, satrec.xli, mm, satrec.xni,
nodem, dndt, nm
);
} // if method = d
//
//
if (nm <= 0.0)
{
printf("# error nm %f\n", nm);
satrec.error = 2;
}
am = pow((xke / nm),x2o3) * tempa * tempa;
nm = xke / pow(am, 1.5);
em = em - tempe;
// fix tolerance for error recognition
if ((em >= 1.0) || (em < -0.001) || (am < 0.95))
{
printf("# error em %f\n", em);
satrec.error = 1;
}
if (em < 0.0)
em = 1.0e-6;
mm
= mm + satrec.no * templ;
xlm
= mm + argpm + nodem;
emsq
= em * em;
temp
= 1.0 - emsq;
nodem
argpm
xlm
mm
=
=
=
=
fmod(nodem, twopi);
fmod(argpm, twopi);
fmod(xlm, twopi);
fmod(xlm - argpm - nodem, twopi);
/* ----------------- compute extra mean quantities ------------- */
sinim = sin(inclm);
cosim = cos(inclm);
//
/* -------------------- add lunar-solar periodics -------------- */
ep
= em;
xincp = inclm;
argpp = argpm;
nodep = nodem;
mp
= mm;
sinip = sinim;
cosip = cosim;
{
dpper
(
satrec.e3,
satrec.ee2, satrec.peo,
satrec.pgho, satrec.pho, satrec.pinco,
satrec.plo, satrec.se2, satrec.se3,
satrec.sgh2, satrec.sgh3, satrec.sgh4,
satrec.sh2, satrec.sh3, satrec.si2,
satrec.si3, satrec.sl2, satrec.sl3,
satrec.sl4, satrec.t,
satrec.xgh2,
satrec.xgh3, satrec.xgh4, satrec.xh2,
satrec.xh3, satrec.xi2, satrec.xi3,
satrec.xl2, satrec.xl3, satrec.xl4,
satrec.zmol, satrec.zmos, satrec.inclo,
'n', ep, xincp, nodep, argpp, mp
);
if (xincp < 0.0)
{
xincp = -xincp;
nodep = nodep + pi;
argpp = argpp - pi;
}
if ((ep < 0.0 ) || ( ep > 1.0))
{
printf("# error ep %f\n", ep);
satrec.error = 3;
}
} // if method = d
84
/* -------------------- long period periodics ------------------ */
{
sinip = sin(xincp);
cosip = cos(xincp);
satrec.aycof = -0.5*j3oj2*sinip;
// sgp4fix for divide by zero for xincp = 180 deg
if (fabs(cosip+1.0) > 1.5e-12)
satrec.xlcof = -0.25 * j3oj2 * sinip * (3.0 + 5.0 * cosip) / (1.0 + cosip);
else
satrec.xlcof = -0.25 * j3oj2 * sinip * (3.0 + 5.0 * cosip) / temp4;
}
axnl = ep * cos(argpp);
temp = 1.0 / (am * (1.0 - ep * ep));
aynl = ep* sin(argpp) + temp * satrec.aycof;
xl
= mp + argpp + nodep + temp * satrec.xlcof * axnl;
/* --------------------- solve kepler's equation --------------- */
u
= fmod(xl - nodep, twopi);
eo1 = u;
tem5 = 9999.9;
ktr = 1;
//
sgp4fix for kepler iteration
//
the following iteration needs better limits on corrections
while (( fabs(tem5) >= 1.0e-12) && (ktr <= 10) )
{
sineo1 = sin(eo1);
coseo1 = cos(eo1);
tem5
= 1.0 - coseo1 * axnl - sineo1 * aynl;
tem5
= (u - aynl * coseo1 + axnl * sineo1 - eo1) / tem5;
if(fabs(tem5) >= 0.95)
tem5 = tem5 > 0.0 ? 0.95 : -0.95;
eo1
= eo1 + tem5;
ktr = ktr + 1;
}
//
/* ------------- short period preliminary quantities ----------- */
ecose = axnl*coseo1 + aynl*sineo1;
esine = axnl*sineo1 - aynl*coseo1;
el2
= axnl*axnl + aynl*aynl;
pl
= am*(1.0-el2);
if (pl < 0.0)
{
printf("# error pl %f\n", pl);
satrec.error = 4;
}
else
{
rl
= am * (1.0 - ecose);
rdotl = sqrt(am) * esine/rl;
rvdotl = sqrt(pl) / rl;
betal = sqrt(1.0 - el2);
temp
= esine / (1.0 + betal);
sinu
= am / rl * (sineo1 - aynl - axnl * temp);
cosu
= am / rl * (coseo1 - axnl + aynl * temp);
su
= atan2(sinu, cosu);
sin2u = (cosu + cosu) * sinu;
cos2u = 1.0 - 2.0 * sinu * sinu;
temp
= 1.0 / pl;
temp1 = 0.5 * j2 * temp;
temp2 = temp1 * temp;
/* -------------- update for short period periodics ------------ */
{
cosisq
= cosip * cosip;
satrec.con41 = 3.0*cosisq - 1.0;
satrec.x1mth2 = 1.0 - cosisq;
satrec.x7thm1 = 7.0*cosisq - 1.0;
}
mrt
= rl * (1.0 - 1.5 * temp2 * betal * satrec.con41) +
0.5 * temp1 * satrec.x1mth2 * cos2u;
su
= su - 0.25 * temp2 * satrec.x7thm1 * sin2u;
xnode = nodep + 1.5 * temp2 * cosip * sin2u;
xinc = xincp + 1.5 * temp2 * cosip * sinip * cos2u;
mvt
= rdotl - nm * temp1 * satrec.x1mth2 * sin2u / xke;
rvdot = rvdotl + nm * temp1 * (satrec.x1mth2 * cos2u +
1.5 * satrec.con41) / xke;
/* --------------------- orientation vectors ------------------- */
85
sinsu
cossu
snod
cnod
sini
cosi
xmx
xmy
ux
uy
uz
vx
vy
vz
= sin(su);
= cos(su);
= sin(xnode);
= cos(xnode);
= sin(xinc);
= cos(xinc);
= -snod * cosi;
= cnod * cosi;
= xmx * sinsu + cnod *
= xmy * sinsu + snod *
= sini * sinsu;
= xmx * cossu - cnod *
= xmy * cossu - snod *
= sini * cossu;
cossu;
cossu;
sinsu;
sinsu;
/* --------- position and velocity (in km and km/sec) ---------- */
r[0] = (mrt * ux)* radiusearthkm;
r[1] = (mrt * uy)* radiusearthkm;
r[2] = (mrt * uz)* radiusearthkm;
v[0] = (mvt * ux + rvdot * vx) * vkmpersec;
v[1] = (mvt * uy + rvdot * vy) * vkmpersec;
v[2] = (mvt * uz + rvdot * vz) * vkmpersec;
} // if pl > 0
//
// sgp4fix for decaying satellites
if (mrt < 1.0)
{
printf("# decay condition %11.6f \n",mrt);
satrec.error = 6;
}
return satrec.error;
} // end sgp4
86
/* ----------------------------------------------------------------------------*
*
function gstime
*
* this function finds the greenwich sidereal time.
*
* author
: david vallado
719-573-2600
1 mar 2001
*
* inputs
description
range / units
*
jdut1
- julian date in ut1
days from 4713 bc
*
* outputs
:
*
gstime
- greenwich sidereal time
0 to 2pi rad
*
* locals
:
*
temp
- temporary variable for doubles
rad
*
tut1
- julian centuries from the
*
jan 1, 2000 12 h epoch (ut1)
*
* coupling
:
*
none
*
* references
:
*
vallado
2004, 191, eq 3-45
* --------------------------------------------------------------------------- */
double
gstime
(
double jdut1
)
{
const double deg2rad = pi / 180.0;
double
temp, tut1;
tut1 = (jdut1 - 2451545.0) / 36525.0;
temp = -6.2e-6* tut1 * tut1 * tut1 + 0.093104 * tut1 * tut1 +
(876600.0*3600 + 8640184.812866) * tut1 + 67310.54841; // sec
temp = fmod(temp * deg2rad / 240.0, twopi); //360/86400 = 1/240, to deg, to rad
// ------------------------ check quadrants --------------------if (temp < 0.0)
temp += twopi;
}
return temp;
// end gstime
87
/* ----------------------------------------------------------------------------*
*
function getgravconst
*
* this function gets constants for the propagator. note that mu is identified to
*
facilitiate comparisons with newer models.
*
* author
: david vallado
719-573-2600
21 jul 2006
*
* inputs
:
*
whichconst - which set of constants to use 72, 84
*
* outputs
:
*
tumin
- minutes in one time unit
*
radiusearthkm - radius of the earth in km
*
xke
- reciprocal of tumin
*
j2, j3, j4 - un-normalized zonal harmonic values
*
j3oj2
- j3 divided by j2
*
* locals
:
*
mu
- earth gravitational parameter
*
* coupling
:
*
none
*
* references
:
*
norad spacetrack report #3
*
--------------------------------------------------------------------------- */
void getgravconst
(
double& tumin,
double& radiusearthkm,
double& xke,
double& j2,
double& j3,
double& j4,
double& j3oj2
)
{
double mu;
switch (whichconst)
{
// -- wgs-72 low precision str#3 constants -case wgs72old:
radiusearthkm = 6378.135;
// km
xke
= 0.0743669161;
tumin = 1.0 / xke;
j2
=
0.001082616;
j3
= -0.00000253881;
j4
= -0.00000165597;
j3oj2 = j3 / j2;
break;
// ------------ wgs-72 constants -----------case wgs72:
mu
= 398600.8;
// in km3 / s2
// km
xke
= 60.0 / sqrt(radiusearthkm*radiusearthkm*radiusearthkm/mu);
tumin = 1.0 / xke;
j2
=
0.001082616;
j3
= -0.00000253881;
j4
= -0.00000165597;
j3oj2 = j3 / j2;
break;
case wgs84:
// ------------ wgs-84 constants -----------mu
= 398600.5;
// in km3 / s2
// km
xke
= 60.0 / sqrt(radiusearthkm*radiusearthkm*radiusearthkm/mu);
tumin = 1.0 / xke;
j2
=
0.00108262998905;
j3
= -0.00000253215306;
j4
= -0.00000161098761;
j3oj2 = j3 / j2;
break;
default:
fprintf(stderr,"unknown gravity option (%d)\n",whichconst);
break;
}
}
// end getgravconst
88

Revisiting Spacetrack Report #3

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